Parallel to the Heidelberg Laureate Forum, the Heidelberg Laureate Forum Foundation organizes an exhibition related to mathematics or computer science, as part of the accompanying program of the HLF. In 2019 it was the mathematics …

Der Beitrag Aldo Spizzichino exhibition at the HLF 2017 erschien zuerst auf Heidelberg Laureate Forum.

]]>Parallel to the Heidelberg Laureate Forum, the Heidelberg Laureate Forum Foundation organizes an exhibition related to mathematics or computer science, as part of the accompanying program of the HLF. In 2019 it was the mathematics of music, in 2018 it was the turn of women in Mathematics. While we wait to know what the exhibition will be in the next edition of the HLF in September 2020, we review the exhibition hosted in the 5th HLF that took place in September 2017, with the work of Italian physicist Aldo Spizzichino on computer-generated art.

For more about Aldo Spizzichino’s work, visit his website,.

Der Beitrag Aldo Spizzichino exhibition at the HLF 2017 erschien zuerst auf Heidelberg Laureate Forum.

]]>I have a strange relationship with my calculator. A lot of people who haven’t met many mathematicians assume when they meet me that my status as a mathematician qualifies me as a super-brain who would …

Der Beitrag Calculating Percentages erschien zuerst auf Heidelberg Laureate Forum.

]]>I have a strange relationship with my calculator. A lot of people who haven’t met many mathematicians assume when they meet me that my status as a mathematician qualifies me as a super-brain who would never resort to working something out using a calculator – once, on being introduced to someone who’d had a few drinks, I was greeted with “Oh, so you’re a mathematician, eh? Then what’s the answer to [mathematical calculation]?” – missing entirely what maths is actually all about – and also thankfully having drunk enough not to be able to come up with a question that was actually difficult. (It turns out, just because numbers are big, it doesn’t mean they’re hard to multiply together, and seven times a million is just seven million).

Being a mathematician is obviously not just all about adding and multiplying numbers. As I hope I’ve been getting across in this column, maths is a rich and beautiful subject full of abstraction and variety, and involves manipulating complicated concepts in your mind to appreciate their beauty. However, it’s unavoidable that mathematicians do sometimes have to do calculations. Occasionally, it’ll be part of work – but most of the time it’s for one of the many mundane reasons that any person has to do a calculation, like working out how much of something you need, or adding up the bill in a restaurant and including a tip. Quite often in these situations, if I want to make sure a small mistake doesn’t result in me being short-changed (or short-changing someone else), I’ll whip out a calculator to save time and effort.

It was on such an occasion recently that I discovered an interesting quirk of how the calculator I was using (the calculator app, on my Android phone) works. I needed to work out the total of my food, plus a 10% service charge – so I added together my main, side and drink, and then calculated that total with 10% added. Normally, I’d do this by typing:

(Here multiplying by 1.1 gives me 110% of the original value, so I’ve added 10%). But I noticed that my friend nearby, who was undertaking a similar process, was using the % key on the calculator. As someone used to converting in my head between percentages and decimals, I’d never really used that key much – but it turns out, Android interprets it in a very natural way. You can type:

And it’ll give the same calculation. What’s more interesting is, you can do this without the brackets, and it’ll still apply it to the whole thing:

As someone who’s involved in both mathematics and maths teaching, this grated on me a little. I appreciate that this is how the average calculator user might want this to work – there aren’t many situations where you would specifically want the percentage only to be applied to the last entry in the sum, so it makes sense that doing this would work in this way. But in my mind, the operation ‘add 10%’ is interchangeable with ‘multiply by 1.1’ – and under the standard operator precedence rules, the multiplication should be calculated before the addition. Indeed,

gives a different answer. I can force the calculator to work it out this way – for instance,

will do the thing I would expect it to do. But it’s intriguing that the calculator takes a natural, but slightly non-mathematical approach here. Since I can get a rough idea of what the answer should be in my head (the total is going to be around 18-19, so adding a 10% tip should make it around or slightly more than 20, since that’ll be around 1.8-1.9), I can usually tell which of these things the calculator has done, but someone relying solely on their calculator to give them the answer is unlikely to notice.

It hasn’t always been this way – stock apps like the calculator are updated all the time, and with new versions of the operating system they can change their behaviour. Previous versions of Android included a calculator app which would interpret ‘10%’ just to mean 0.1 (which is precisely what it means, and much closer to my mathematical interpretation of the calculation). So you’d get

It’s taken the sum of the three numbers, then added 0.1. Unhelpful! Under this interpretation of what % means, to work out my bill I could just multiply by 110%, which the calculator would treat as multiplying by 1.1 as I’d usually do. But would it make sense to someone who doesn’t think of it that way?

Writing a calculator app can be a big challenge. Phones can perform incredible feats of processing – streaming video, running games and presenting huge complex websites, coordinating multiple input and output devices, all on something that fits in your pocket. So you’d think that to make something that you operate by pressing buttons on a screen, which output the answer to a mathematical calculation – something computers do all the time – would be trivial.

But calculators have always been an interesting challenge for programmers. The way computers store and process numbers often causes hiccups and issues with doing precise calculations. For instance, numbers are stored to a particular degree of precision – a fixed number of total decimal places. Called ‘floating-point arithmetic’, it involves storing a fixed number of digits, as well as recording the position of the decimal point (using a power of the base you’re calculating in, just like the standard scientific form of [number] × 10^{[power]}). So if you’re calculating with very big, or very precise values, sometimes rounding occurs, and can cause errors.

Another issue is with certain numbers which behave differently in binary. Some numbers in decimal can’t be stored using a finite string – for example, 2/3 = 0.666666 (recurring forever) – so it’ll be stored by a calculator with a finite memory as 0.666666667 (for a given size of memory). When computers convert numbers to binary, they can still store non-whole numbers – for example, ½ in binary is 0.1. The columns which are numbered 1,2,4,8… going off to the left are just 2^{0}, 2^{1}, 2^{2}, 2^{3}… – so the columns off to the right are 2^{-1}, 2^{-2}, 2^{-3} and so on, giving ½, 1/2^{2}, 1/2^{3}… In this way, ¼ is 0.01 and ⅛ is 0.001.

This means that certain numbers can’t be stored in binary using a finite string of digits either – and it’s a different set of numbers than it is in decimal. For example, 0.34 has a finite decimal expansion, but in binary it’s 0.0[10101110000101000111], where the section in brackets repeats forever. So this number, along with infinitely many others, can’t be stored exactly in binary, meaning errors can creep in if not handled carefully.

But as well as all these technical challenges, designers of calculator apps also have to consider the way people will use the calculators. Given a concept like the percentage button, there are any number of ways to interpret what the user could mean when they use it. This seems strange, since it’s a specific mathematical concept, and ideas in maths are usually well-defined – but it’s also a concept in general use, so it needs to work the way most normal people (one way to define non-mathematicians) expect it to.

Another popular type of calculator people carry around in their pockets is the iPhone – and the built-in calculator there actually behaves differently. If you type in a calculation like 12.50+2.95+3.5+10%, as soon as you press the percent symbol, it’ll calculate and display the percentage – in this case, 10% of (12.50+2.95+3.5). If you then proceed to press equals, it’ll add this to the total, and show you the correct value.

This is equivalent to what the Android calculator does – it’s still giving you the answer you need so you can hand the server the correct amount of money – but it’s at least acknowledging that there’s an intermediate step, so you can see a bit of what’s going on.

Perhaps this strange percentage handling is a function of calculators designed for general use. Scientific calculators, made for engineers and mathematicians to do precise and complex calculations, handle the percentage key very differently. I’ve been playing with the scientific calculator I used for my school exams, and it gives some very strange answers when I try to add a percentage – multiplying by one is fine, as you would expect, but it turns out that even different calculator manufacturers have implemented wildly differing interpretations of what ‘18.95 + 10%’ means, and it took me a few minutes to work out what my Casio was even doing.

I suppose the real lessons here are, 1. Don’t take your scientific calculator to a restaurant, 2. Make sure you know what you’re asking a calculator to do so you can check it’s doing the right thing, and 3. Always tip your server (the right amount).

Der Beitrag Calculating Percentages erschien zuerst auf Heidelberg Laureate Forum.

]]>A friend of mine recently kindly made me a present, using their 3D printer – a puzzle made up of 21 pieces, three each of seven different shapes. The statement of the puzzle is as …

Der Beitrag Building Blocks erschien zuerst auf Heidelberg Laureate Forum.

]]>A friend of mine recently kindly made me a present, using their 3D printer – a puzzle made up of 21 pieces, three each of seven different shapes. The statement of the puzzle is as follows:

“Can you arrange the pieces into seven groups of three so that for each possible pair of shapes, there is one group containing that pair?”

That is, I need to find a way to arrange the 21 pieces into 7 groups, so that if you name any two shapes, I can point to a group (and only one group) which contains those two pieces. The pieces are mounted on bases shaped like thirds of a circle, so groups of three fit together nicely, and we’ve had a lot of fun coming up with names for the seven shapes.

It’s an interesting puzzle, and it took me a few minutes to work out a solution – mostly by trial and error. I first noticed that it didn’t matter which three shapes I picked to put in the first group, since there are the same number of each shape, so they’re interchangeable. Then I tried some other groupings alongside that, and worked backwards whenever I reached a problem.

If you’d like to have a go at the puzzle, I’ve made a PDF of 21 pieces you can cut up, with three each of seven different shapes – you’ll need access to a 2D printer and a pair of scissors.

One way to extend this puzzle is to try it without paying attention to the colours first, then once you have a solution, try to rearrange the pieces so each group has one piece of each colour in it.

This kind of problem – looking at different ways to combine things so that they satisfy statements like ‘any two shapes occur in exactly one group’ is part of a branch of maths called combinatorics. I’ve mentioned it before in a previous blog post, as it also covers counting and enumerating combinations, but this kind of problem constitutes one particular branch – and solutions to them are called combinatorial block designs.

The term ‘block designs’ refers to the fact that it’s about ways of grouping things, in such a way that specific combinations occur. For example, a group of friends planning to play a series of games together might wish to work out a set of groupings they can arrange themselves in so that they all get to play against everyone else:

6 people want to play in three teams of two, for five different rounds of a game; each round, everyone would like to be on a team with someone they’ve not been on a team with before. How can they arrange themselves?

(One solution to this problem is given at the end of the post – but see if you can find one yourself!).

One very old problem in this area, which was originally stated in 1850 and appeared in several recreational mathematics books at the turn of the century, can be stated as follows:

Fifteen people wish to arrange themselves into groups of three, with a different arrangement each day for a week, so that no two people are in the same group on more than one day.

This problem can be solved (and solutions are given on the Wikipedia page) using a **block design**. This is a way of splitting up a set of things, call it X, into a collection of smaller sets – called blocks – with various constraints which can be applied depending on the desired properties – including:

- Uniformity – all the blocks are the same size (the most interesting and most often studied type)
- Any element of X is contained in a specific number of blocks – which may be one, or more than one if the sets can overlap
- Any set of elements from X of a given size is contained in a specific number of blocks

For example, in the 3D printed puzzle I was given, there are 7 blocks of size 3, any one shape is contained in exactly 3 blocks (as there are three pieces of each shape) and for any set of shapes of size 2, there is exactly one block which contains that set.

In the case of the 19th-century maths problem, there are 15 people who need to be arranged into groups across all seven days – so there are five groups of three each day, and each person is in exactly seven groups, one for each day.

Then the constraint is that no pair of people are in the same group more than once. This problem has the additional constraint that the seven groups each person is in all occur on different days – the groups form a partition of the people each day, but the groups themselves can be partitioned into seven days which contain each person once.

In cases where the design needs to have combinations of things occurring exactly once, such block designs are called **Steiner Systems** and are denoted S(t,k,n) where n is the total number of things, k is the size of each block, and t is the size of combination that has to occur exactly once. My puzzle is an S(2,3,7) system – and so is the diagram given here.

This shape is called the **Fano plane**, and is a visual representation of an S(2,3,7) system. It has seven points, which are arranged into lines – each of which passes through exactly three points, and any two points you can choose lie only on one line.

If you want an easy way to solve the seven shapes puzzle, or you’ve solved it and want to compare your solution to this structure, you could arrange each set of three shapes so it sits on one of the seven points – in such a way that each line has the three pieces of that shape sitting along it.

It’s possible to construct Steiner systems of the form S(2, q+1, q²+q+1) – the Fano plane is for the case q=2. There’s a chance you may also have encountered equivalent structures for other values of q – if you’re a fan of children’s card games.

Dobble (sold in some places as Spot-It!) is a simple picture-spotting game, which comes in a bright yellow and purple tin. It consists of a set of circular cards, each of which is covered in pictures of familiar objects (a car, a key, a turtle, etc.) – eight objects appear on each card, and there are various ways to play, but all involve comparing two cards and finding an object which appears on both cards.

Of course, if this game weren’t mathematically interesting, I wouldn’t be talking about it – and in fact, for any pair of cards in the game, there will always be a symbol, and exactly one symbol, which appears on both cards. In a way which has no impact whatsoever on children’s enjoyment of spotting and matching and shouting out the name of the symbol they’ve found in common, the game has a beautiful underlying mathematical structure.

Since there are 8 symbols on each card, and 57 symbols in total, it’s clear that this is an S(2,8,57) system (S(2, q+1, q²+q+1) where q=7). It also means that each symbol occurs on exactly 8 of the cards, and a quick rifle through the set I have confirms that there are only 8 cards with the turtle on.

There are various versions of the game available, including Beach Dobble – with waterproof cards, designed for use at the beach. Since this was clearly designed to be a portable version of the game, it’s got a smaller deck – but it’s still a Steiner system! Each card has 6 symbols on it, so we must have q=5; and indeed, there are 5²+5+1 = 31 different symbols, with each symbol appearing on 6 cards.

Since the whole thing is symmetrical, it’s also possible to construct 31 cards to go in this set – and in standard Dobble, there are 57 possible cards you could construct using the 57 symbols. However, for reasons unknown to mathematics, and presumably related to printing constraints, the makers of Dobble have decided to only include 55 cards in the game! Beach Dobble, with 31 possible cards, only includes 30.

This means there are two more combinations in standard Dobble, and one in Beach Dobble, which are missing from the set. Perhaps they wanted to give mathematicians the gift of an additional puzzle – to find which cards they are missing. If you’ll excuse me, I’m going to find count how many occurrences of each symbol there are on the cards, determine what the missing ones should be and design and print my own replacements. Hours of fun for all the family!

- Read Christian’s blog post about 3D printing the puzzle, including a link to 3D printing files you can use to print your own.
- One solution to the ‘six people, three pairs five ways’ puzzle from the top of the post: AB CD EF / AC BE DF / AD BF CE / AE BD CF / AF BC DE

Der Beitrag Building Blocks erschien zuerst auf Heidelberg Laureate Forum.

]]>Often at this time of year, people are visiting family, and sometimes traditional games and toys come out to occupy your time. If you’ve got a deck of playing cards handy, you can perform a …

Der Beitrag A mathematical card trick erschien zuerst auf Heidelberg Laureate Forum.

]]>Often at this time of year, people are visiting family, and sometimes traditional games and toys come out to occupy your time. If you’ve got a deck of playing cards handy, you can perform a trick to amuse people – and since it’s a mathematical card trick, you don’t need to be a magician to make it work!

- Count out 27 cards from your deck, and put the rest away.
- Ask a volunteer to shuffle the 27 cards, pick one, look at it, and return it to the stack (in its original position, or anywhere they like) – all without you seeing the card.
- Deal the 27 cards into three face-up piles (placing one card in each pile, then repeating this 9 times). Ask your volunteer to watch for the card they picked – once you’ve finished dealing all the cards, they should tell you which pile it’s in.
- Pick up the three piles, and return them to a stack of 27 cards, but making sure you place the stack your volunteer indicated
**in between the other two**. - Repeat steps 3 and 4 two more times, so you’ve done the process three times in total.
- Count cards off from the top of the deck and turn over the 14th card (using the excuse that 13 is an unlucky number, so 14 must be a lucky number) – this should be your volunteer’s card.

You can try this trick and make sure it works – if it goes wrong, it’s usually a miscount (check you actually have 27 cards!), or a step somewhere that’s gone wrong, so try again or ask a friend to watch and make sure you’re doing it right.

Given that you know this is a mathematical card trick, it should make sense that as long as all the steps are followed correctly, you end up with the same result every time. But this trick has some interesting underlying mathematics, which can be used to perform an even more impressive version of the trick. First we need to understand what’s happened in our 14th-card version of the trick. Imagine your volunteer indicates the card is in the left pile.

When the cards are re-stacked, the pile containing your volunteer’s chosen card is placed in the middle. This means that when you deal the second time, you know that the card you’re looking for isn’t in the first nine cards, or the last nine cards. Since we deal the cards across the three piles, this means your chosen card won’t be in the bottom three or top three cards of any of the three piles. (Think about this a bit more if you’re not convinced.)

Now imagine your volunteer indicates the right-hand pile. This tells you which set of three cards (from the middles of the three piles) the card is in – and this set of three, if you place the pile in the middle again, will spread itself out as the middle card in each of the three piles on the third deal.

If the pile pointed out to you the third time is then placed in the middle, the card will be the middle card in the middle pile of three, or in other words, in the middle of the whole stack – which makes it the 14th card. Magic!

So the trick works, and it makes sense. True mathematicians will at this point become curious – we placed the pile in the middle each time, but we didn’t have to do that. It could equally have been placed on the top or bottom, on each of the three deals. So what would happen if we did that?

A little thought, and running through the trick with different combinations, reveals the following observations:

- Placing the chosen pile on the top each of the three times makes the card finish on top of the deck
- Placing the chosen pile on the bottom each time makes the card finish at the bottom.

With a bit more investigation, the following observations become apparent:

- Placing the pile on top after the third deal will always result in it ending up in the top third of the deck; similarly placing it in the middle third means it’s in the middle third of the deck, and the same for the bottom third
- Within each third, the position of the pile after the second deal will determine which third of that third it’s in (top, middle or bottom)
- Within that third-of-a-third, the position of the pile after the first deal determines whether it’s the top, middle or bottom of those three cards.

This information can be summarised in a table – here the ‘final position’ is the number of cards that will be on top of the chosen card at the end, with 0 cards on top of the top card, and 26 cards on top of the bottom card.

First deal | Second deal | Third deal | Final position |

top | top | top | 0 |

middle | top | top | 1 |

bottom | top | top | 2 |

top | middle | top | 3 |

middle | middle | top | 4 |

bottom | middle | top | 5 |

top | bottom | top | 6 |

middle | bottom | top | 7 |

bottom | bottom | top | 8 |

top | top | middle | 9 |

middle | top | middle | 10 |

bottom | top | middle | 11 |

top | middle | middle | 12 |

middle | middle | middle | 13 |

bottom | middle | middle | 14 |

top | bottom | middle | 15 |

middle | bottom | middle | 16 |

bottom | bottom | middle | 17 |

top | top | bottom | 18 |

middle | top | bottom | 19 |

bottom | top | bottom | 20 |

top | middle | bottom | 21 |

middle | middle | bottom | 22 |

bottom | middle | bottom | 23 |

top | bottom | bottom | 24 |

middle | bottom | bottom | 25 |

bottom | bottom | bottom | 26 |

This means we have a pattern we can use for finding the combinations of top, middle and bottom we need to put the card in a specific place at the end of the trick. It might seem a little intimidating to memorise all these combinations, but of course mathematics has the answer.

If you’re familiar with binary numbers, you’ll know that it’s possible to represent any number using only ones and zeroes – base 2, which means every digit is one of two options. But if we wanted three options, that’s also possible – using base 3. Ternary numbers, as they’re known, are a way to write any number as a sum of powers of three, and each digit is either 0, 1 or 2.

The table below shows how to write numbers from 0 to 26 in ternary – and it bears a striking resemblance to our other table above – replace 0 with ‘top’, 1 with ‘middle’ and 2 with ‘bottom’ and you have the same table! The columns represent how many 1s, 3s and 9s (reading from right to left) are added together to make each number – for example, 7 = 3+3+1 so it’s written 021, and 12 = 9+3 so it’s written 110.

1s | 3s | 9s | Total |

0 | 0 | 0 | 0 |

1 | 0 | 0 | 1 |

2 | 0 | 0 | 2 |

0 | 1 | 0 | 3 |

1 | 1 | 0 | 4 |

2 | 1 | 0 | 5 |

0 | 2 | 0 | 6 |

1 | 2 | 0 | 7 |

2 | 2 | 0 | 8 |

0 | 0 | 1 | 9 |

1 | 0 | 1 | 10 |

2 | 0 | 1 | 11 |

0 | 1 | 1 | 12 |

1 | 1 | 1 | 13 |

2 | 1 | 1 | 14 |

0 | 2 | 1 | 15 |

1 | 2 | 1 | 16 |

2 | 2 | 1 | 17 |

0 | 0 | 2 | 18 |

1 | 0 | 2 | 19 |

2 | 0 | 2 | 20 |

0 | 1 | 2 | 21 |

1 | 1 | 2 | 22 |

2 | 1 | 2 | 23 |

0 | 2 | 2 | 24 |

1 | 2 | 2 | 25 |

2 | 2 | 2 | 26 |

We can decide the position we want the card to be in at the end of the trick, look it up in the table, and get a pattern of top, middle and bottom (0, 1 and 2) to put the card in the right position.

As we’ve seen, 111 (middle middle middle) gives 9+3+1=13 cards on top, putting it as the 14th card. 201 (bottom top middle) gives 9+9+1=19 cards on top, putting it as the 20th card. In fact, you could start the trick by asking someone to pick a card, and also casually ask them their age (if you think it’s likely to be less than 27, and not cause offence!) or to pick their favourite number between 1 and 27 – then that would be the number you count to at the end of the trick.

If you put the piles together casually enough, nobody will notice you’re stacking them in a particular order, and the effect can be very impressive. Mathematical magic tricks like this are favourites among professional magicians, as they don’t require any sleight of hand, and always work correctly if you get all the steps right. I personally like this trick because it’s a wonderful application of a nice bit of mathematics!

Der Beitrag A mathematical card trick erschien zuerst auf Heidelberg Laureate Forum.

]]>Here are the solutions to the eight candle puzzles we posted earlier. The candles cost €1.05, and the matches cost €0.05. Every 30 seconds the number of people whose candles are lit doubles, starting with …

Der Beitrag Candle puzzles – solutions erschien zuerst auf Heidelberg Laureate Forum.

]]>Here are the solutions to the eight candle puzzles we posted earlier.

- The candles cost €1.05, and the matches cost €0.05.

- Every 30 seconds the number of people whose candles are lit doubles, starting with 1 person at time 0. After 30 seconds, 2 = 21 candles are lit, after 60 seconds 4 = 22 candles are lit, and so on. After 7 30-second gaps, 27 = 128 candles will be burning, so it will take until 8 lots of 30 seconds (4 minutes) have passed for all the candles to be lit.

- This can be achieved by lighting one of the candles at both ends, and allowing it to burn from both ends at the same time – the flames will meet in the middle after 30 minutes, at which point you can light the second candle and let it burn for an hour. (This does assume that the candles burn at the same rate regardless of which way up they are – in reality, if a candle was held upright and lit at both ends, the bottom end would burn more quickly. If you hold the candle horizontally it might also burn at a different rate than it burns from the top if held upright. But it’s a nice puzzle!)

- If you place three of the candles on each side of the scale, this will indicate which group of three the heavy candle is in: if the scale tips one way or the other, it’s in that group of three, and if it balances exactly, it’s in the three candles you didn’t use for this weighing. Take the group of three that the heavy candle is in, and place one candle on each side of the scale – then the same principle will apply to tell you which is the heavy one.

- After burning 100 candles you’ll have 10 candles’ worth of extra wax; this lets you make 10 more candles, each of which will yield another 1/10 of a candle of wax – so you could make 11 extra candles in total.

- If you blow on each candle once, starting with the one nearest to you and going round one at a time, when you get back to the start you’ll have blown out all the candles! This is because they each get blown on three times (three candles change from lit to unlit or vice versa on each blow), which is an odd number of times, so they’ll go from lit to unlit eventually.

- The number of candles you will have blown out is the sum of all the ages you’ve been – when you’re five you’ll have blown ou 1+2+3+4+5 = 15 candles. These are triangular numbers, and the general pattern is N×(N+1) / 2.

- If you have five 1s and five 0s, the first time you’ll hit a problem is when you need six of the same digit – which will happen for the first time when you try to write the number 111111 = 1+2+4+8+16+32 = 63.

Der Beitrag Candle puzzles – solutions erschien zuerst auf Heidelberg Laureate Forum.

]]>Around this time of year, people light a lot of candles in celebration of the season. Here is a selection of candle-based maths puzzles you can share with friends and family! We’ll share the solutions …

Der Beitrag Eight Candle-based Maths Puzzles erschien zuerst auf Heidelberg Laureate Forum.

]]>Around this time of year, people light a lot of candles in celebration of the season. Here is a selection of candle-based maths puzzles you can share with friends and family! We’ll share the solutions in a later post.

- You go to the shop to buy a box of candles and a box of matches. The total cost is €1.10, and the shopkeeper says that the candles cost €1 more than the matches. How much did the matches cost?

- 200 people are gathered in a room for a party, and everyone has an unlit candle. One person lights their candle, and then walks to another person in the room whose candle is not yet lit, and uses it to light that person’s candle. Then both people whose candles are burning go and find someone else’s candle to light, and so on – at each stage, anyone whose candle is already lit finds one other person and lights their candle. If it takes 30 seconds for each person (including the first) to find a friend and light a candle, how long does it take from when the first candle is lit to when all the candles in the room are burning?

- You have two candles which each take 1 hour to fully burn, and they burn at a constant rate along their length. You also have a lighter. You’re given the task of measuring exactly 90 minutes using the objects you have (and nothing else – no clock, weighing scale or cutting equipment). How can you do this?

- You have nine identical candles, but one is made from a different type of wax which means it is heavier than the others. You have only a balance scale to compare the weights of objects. What’s the fewest times you need to use the balance to work out which candle is the heavy one?

- The end pieces of wax left over after burning ten candles will yield one extra candle if you melt them all together. If you burned 100 candles, how many extra candles could you make (assuming you have as much wick as you need)?

- Assuming you follow the standard pattern of blowing out the same number of candles on your cake as your age each birthday, how many birthday candles will you blow out in your lifetime, if you live to be N years old?

- It’s your seventh birthday, and in front of you is a round cake with seven candles arranged in a circle around the edge. Your goal is to blow out all of the seven candles – but you’re not very good at directing your breath (you are seven, after all – you haven’t had much practice at blowing out candles). If you blow out one candle, the two candles either side are also blown out. But your friends have set up the cake with trick candles! When they are blown on, if they’re lit they go out, but if they’re currently unlit, they light up again. Whichever candle you blow on, that one and the two candles either side change from lit to unlit depending on what they previously were. How can you blow out all the candles using the fewest breaths?

- I like using number-shaped candles on birthday cakes, but I only have five 0-shaped candles and five 1-shaped candles. If I represent numbers in binary, what’s the smallest age I won’t be able to represent on a cake?

Der Beitrag Eight Candle-based Maths Puzzles erschien zuerst auf Heidelberg Laureate Forum.

]]>You might enjoy playing games with a friend, or a group of friends. The good news is that even if you don’t have any friends, or even if you don’t exist yourself, you can still …

Der Beitrag Zero-player Games erschien zuerst auf Heidelberg Laureate Forum.

]]>You might enjoy playing games with a friend, or a group of friends. The good news is that even if you don’t have any friends, or even if you don’t exist yourself, you can still play mathematical games. Zero-player games are ones which have no sentient players, and the rules of the game are such that it can proceed without input from any human.

There are a few different ways in which a game can have zero players – and not all of them require you to build a functioning game-playing robot before you start.

Cellular automata are mathematically determined systems which are based on cells – conventionally arranged in a grid, but automata can also be defined in more or fewer than two dimensions. The idea is that for each cell, there are a given number of states it can be in (usually indicated by the colour of the cell), and a fixed rule about how the state of each cell changes with time.

Such systems can be thought of as games, in the sense that changes happen to the cells sequentially and can be seen as taking turns. The rules of the game being specified, it can run without any input from a player, since what happens at each stage is determined by the rules. The classic example of a cellular automaton is John Conway’s Game of Life.

Invented in 1970, the Game of Life involves a two-dimensional grid in which cells can either be black or white (in the game, the two states are known as ‘alive’ and ‘dead’) and at each time step, whether a cell is going to be alive or dead for the next time step is determined by how many of its eight immediate neighbours are currently alive. The standard rules are given as:

- Any live cell with fewer than two live neighbours dies, as if by underpopulation.
- Any live cell with two or three live neighbours lives on to the next generation.
- Any live cell with more than three live neighbours dies, as if by overpopulation.
- Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

Given this simple set of rules, and a plain black and white grid, some surprisingly interesting structures arise. Study of this game has largely been devoted to finding sets of cells which, if the game is started with exactly those cells alive, it will continue in particular ways. For example, structures have been found which are stable – they don’t change; others which form a loop of two or three states which repeat indefinitely unless disturbed; and even structures which evolve in such a way that they destroy themselves while producing a copy of themselves one square further across – by which means they move slowly across the grid. Some examples are shown here.

Beehive shape – static Blinker – oscillates between two states Glider – moves diagonally

In fact, it’s been shown that within the Game of Life you can construct systems which can be used to perform any calculation that can be written down as an algorithm – known as Turing complete.

There are plenty of other examples of cellular automata – all possible 1-dimensional automata with two possible states have been enumerated by Stephen Wolfram. These consist of a single row of cells, whose states determine the states of cells in the next row, producing a pattern which can be so beautiful that people will put it on a train station.

Other automata are defined in different ways – for example, Langton’s Ant is an example of an automata which is defined using an ant character on the grid, which moves around changing the states of the cells as it goes according to fixed rules.

Another way to make a game zero-player is to teach a computer how to play it. Ever since computers came into widespread use, people have been trying to teach them how to play games – it’s an interesting programming challenge, and while some games are simple to model, others take huge amounts of computation power.

EDSAC (the Electronic Delay Storage Automatic Calculator) was an early computer which was taught to play noughts and crosses in 1952 – called OXO, the output from the game was displayed on a cathode ray tube, and the input was via a rotary telephone dial! It was widely thought of as the first computer game. Since then, almost all simple games have been modelled and studied using algorithms, and people have also had a go at the more complex ones too.

Chess computers were a huge focus of artificial intelligence development in the 1980s and 90s, since it’s a game which has a simple specified set of rules and objectives, but whose complexity becomes very large very quickly. The number of possible moves in any given situation can be quite large, and checking all the many avenues of possible moves for yourself and your opponent through to the end of the game is too much even for a computer. This forces people writing chess computer algorithms to think about other ways to optimise their strategy – thinking a few moves ahead, and assuming your opponent will choose good moves.

Chess computers surpassed humans in the 1990s – when Deep Blue famously played and beat chess grandmaster Gary Kasparov – but for other, more complex games, like the ancient Japanese game of Go, the number of possibilities is too big even for advanced computers. It was only in 2015 when computers employing machine learning were able to first beat a human at the game. AlphaGo, developed by DeepMind, was trained to play the game partly by playing against itself repeatedly, until it had seen enough possible strategies to learn how to play well. Truly a zero-player game!

There’s one other way in which games can be considered to be zero-player – and there’s a good chance that you’ve played a zero-player games yourself. Many games are considered ‘solved’ – that is, they’re simple enough that they have been fully analysed and all possible strategies are known. One example is noughts and crosses – there are only 31,896 possible game states, and many optimal strategies are known. So if both players are playing optimally, their moves are effectively chosen for them – so there’s not really any input from human players.

Some zero-player games can really feel like you’re actually playing them, when in fact you aren’t. Snakes and ladders is a nice example of this – you take it in turns to roll a die and move your piece, and if you happen to land on a snake or a ladder, you’ll go to a different part of the board. But in a game like this, literally all your movements are determined by the roll of the dice. Maybe you feel like you have some influence on the results of the dice roll – but if it’s a truly fair dice, it shouldn’t be possible, and any throw is equally probable.

So if you’re playing snakes and ladders, you’re not really putting much in. In fact, I’m playing snakes and ladders right now, and have been for a few years – some friends and I wrote a program which rolls a dice and moves us around on an imaginary board, which plays about 1000 games a week and emails us to let us know who’s winning. I’m currently in the lead!

Der Beitrag Zero-player Games erschien zuerst auf Heidelberg Laureate Forum.

]]>Squares – shapes with four right-angle corners, and four equal-length sides – are lovely. Mathematicians especially love them, as evidenced by the number of nice geometrical and numerical constructions based on squares. For example, a …

Der Beitrag Perfect Squares erschien zuerst auf Heidelberg Laureate Forum.

]]>Squares – shapes with four right-angle corners, and four equal-length sides – are lovely. Mathematicians especially love them, as evidenced by the number of nice geometrical and numerical constructions based on squares. For example, a square number is one which is made by multiplying a number by itself – and the resulting number, if it were a number of beads, could be laid out in a square grid.

As well as squaring numbers, some mathematicians also like to square shapes. Squaring a shape involves dividing it up exactly into squares with whole number (integer) length sides. For example, if I have a 10 by 20 rectangle, I could split it up into two squares of size 10 by 10.

This is pretty simple – but squarings of a shape (ways of covering it by integer-sized squares) can get more complicated. This is a 15-by-25 rectangle covered using squares of size 15, 10, 5 and 5.

For some shapes, this is easy to do, and for others it’s impossible. We can make it into a harder puzzle by adding some extra constraints. For example, a squaring is called **simple** if it has no subset of two or more squares that themselves form a square or rectangle. If such a subset does exist, the squaring is called **compound**.

For example, the squaring above using 15, 10, 5 and 5 can be split to the right of the 15 square, giving two separate pieces that are themselves rectangles. Or I could just break off the two 5s to get a rectangle. If no rectangular piece made of two or more squares can be sectioned off in this way, the squaring is simple.

Here’s a simple squaring of a 15 by 11 rectangle, using squares of size 6, 6, 5, 5, 4, 4, 3, 1 and 1:

If we want to ramp up the difficulty, we could require a squaring to be **perfect** – that is, each of the individual squares making it up needs to be a different size. Such squarings are difficult to find, and it’s a nice puzzle to fit the pieces together. Below is a 33 by 32 rectangle, which has a simple, perfect squaring using squares of size 18, 15, 14, 10, 9, 8, 7, 4 and 1. Can you fit them together to fill the square? One solution is at the bottom of the post.

While squaring rectangles might be fun, it’s still not the most difficult this problem gets. Hardcore squarers are looking to square the ultimate prize: a square.

Dividing up an integer-sized square into distinct integer-sized squares is no mean feat. Squarings of squares and rectangles have been studied since as long ago as 1903, and perfect squarings of squares were first seriously attempted by a group of Cambridge mathematicians in the 1930s. In 1939 the first perfect squared square was published – made up of 55 squares, and measuring 4205 along each side (but sadly a compound squaring). It’s pretty impressive, given that at the time, there were no computers to run searches – these early squares were discovered by hand.

As time progressed and technology caught up with squarers’ ambitions, it was possible to find and confirm the smallest possible perfect squared squares using computer programs. The fewest possible squares a perfect squared square can be squared with is 21, and such a squaring was found in 1978 by A. J. W. Duijvestijn. It measures 112 along each side, and is now used as the logo for the maths society at Trinity College, Cambridge.

However, it’s not the actual smallest a squared square can be – that honour is held by three different squares, each comprising 22 squares and measuring 110 along the side. These were proven to be the smallest such squares in 1999. One is shown here.

Even the simplest perfect squared squares are sufficiently unwieldy that they probably aren’t easy to work out by hand, and this problem has benefitted hugely from the application of computer programmes. Mathematicians continue to find new ways to constrain the problem – could we find a perfect squared square with a square positioned exactly symmetrically in the centre? Can we place other constraints on the relative sizes of the squares? What about cubing the cube? The possibilities are endless (and outlined in detail, of course, in an online catalogue at squaring.net).

One of my favourite things about simple perfect squared squares is that they’re so aesthetically pleasing. They don’t have any fault lines or subsections you can easily take out, and the squares jostle together in a way that shouldn’t work, but somehow just exactly does. This might be the reason why they’re used for artworks and decoration – the images below show an inlaid wooden table top, a stained glass window, and a set of coasters, all employing squared square designs.

And finally, here’s the solution to the puzzle above (rotations and reflections of this also work!):

Der Beitrag Perfect Squares erschien zuerst auf Heidelberg Laureate Forum.

]]>Der Beitrag Podcast Interview with “Father of the Internet” Vint Cerf erschien zuerst auf Heidelberg Laureate Forum.

]]>Vint Cerf, known as one of the “fathers of the internet” and a recipient of the 2004 ACM A.M. Turing Award, was a participating laureate at this year’s Heidelberg Laureate Forum (HLF). Cerf currently acts as a vice president and Chief Internet Evangelist for Google. At HLF, I had an opportunity to sit down with Vint and interview him for the Computing Community Consortium’s (CCC) official podcast, “Catalyzing Computing,” which features interviews with researchers and policymakers about their background and experiences in the computing community.

Prior to our interview he also participated in a press conference where he discussed some of the projects he is currently involved with, as well as other hot topics.

Cerf was asked if he anticipates any potential problems that AI might introduce, either with regards to the Internet or more broadly. Cerf responded that he is most concerned about unexpected, emergent properties that might appear as the volume of AI systems increase and they interact with each other and with humans. He noted that the Internet in particular struggles with large, emergent property effects because it is distributed: software will encounter other software in novel ways, potentially leading to adverse effects.

Cerf was also asked about how differences in individuals can advance science, and specifically what role his hearing loss has played in his career. Cerf dispelled the idea that he got involved in electronic messaging because his wife was deaf for fifty years (she has since received cochlear implants) and because he is hard of hearing. Instead he says his wife got into email because of her book club, though he admits that because of his hearing impairment he jumped on ARPANET as soon as it was available since it was more convenient for him than phone calls, and later he joined places that had, or would let him set up, email technology. Despite not necessarily being his primarily catalyst for scientific exploration, Cerf is very interested in the ways that technology can help people with disabilities or can augment people’s capabilities.

Cerf was also asked about the potential of an “information dark age” if current computing capabilities are lost or become obsolete. Cerf expressed concern for this possibility – as an example of the power of preserving information, he cited the book Team of Rivals by Doris Kearns Goodwin, which explores the history of U.S. President Abraham Lincoln and some of his key cabinet members. This book was possible because of the extensive letters and diaries that each man wrote and which Goodwin was able to use as a resource. If a person’s key historical records only exist as tweets and texts, Cerf fears there may come a time when such a book is not possible due to a dearth of surviving material. Even if files are not lost or destroyed, they may become unreadable to newer file formats, rendering them obsolete.

Building off some of these questions, in this episode of the podcast Cerf discusses net neutrality, how to combat spoofing, the future of Internet connected devices, and the challenges of developing interplanetary Internet.

You can stream the episode in the embedded player below or find it on **iTunes**** | ****Spotify ****| ****Soundcloud**** | ****Stitcher**** | ****Google Play**** | ****Youtube**** | ****Blubrry**** | ****iHeartRadio****. **(note, it may take a few hours for the podcast to become available on your preferred platform).

Der Beitrag Podcast Interview with “Father of the Internet” Vint Cerf erschien zuerst auf Heidelberg Laureate Forum.

]]>Here you can take a look at all the artworks created by the Young Researchers who participated in the Intercultural Science-Art Project. Akinade Mary Oluwabunmi (Nigeria) “Nkan tio ba kọjaa mathematics, o kọjaa eniyan”(Yoruba)“Whatever …

Der Beitrag Intercultural Science-Art Project – The Artworks erschien zuerst auf Heidelberg Laureate Forum.

]]>Here you can take a look at all the artworks created by the Young Researchers who participated in the Intercultural Science-Art Project.

**Akinade Mary Oluwabunmi **(Nigeria)

**“****Nkan tio ba kọjaa mathematics, o kọjaa eniyan****”**(Yoruba)

“Whatever goes beyond mathematics, goes beyond man”

2019

digital

Looking externally at the work, one may be quick to conclude that it’s the map of Nigeria, my country of origin. This image, however, does not include the two rivers which divide the country into three regions, hence visualizing the country as a single unit.

The two lines contained in the image are representations of a few of the numerical simulation results performed on a Lassa fever model which I developed in the course of my present masters’ degree research. Lassa Fever is a highly fatal acute Viral Hemorrhagic fever caused by the Lassa Virus. The first occurrence of this disease was in Nigeria in 1969 and presently, Nigeria is home to over 50% of the total human Lassa fever population.

The yellow line in the image represents the numerical simulation result of the model’s infectious class without the incorporation of any control strategy while the red line represents the numerical simulation result of the model’s infectious class with the incorporation of three different control strategies.

The phrase “Nkan tio ba kọjaa mathematics, o kọjaa eniyan” is written in Yoruba, my native language. The English translation is “Whatever goes beyond mathematics, goes beyond man”. This is to simply state that Mathematics is a tool that helps to proffer solutions to all human problems, provided it’s properly maximized.

**Alexis VanderWilt** (USA)

**“****This is not an Infectuous Relationship.****”**(English)

2019

digital

My research focuses on the impact of social networks on the spread of disease. So, each dot represents a person and their color represents their status whether it be Infected, Dead, Recovered, or Susceptible. The size of each dot represents how social they are. The hands represent one of the many interactions between an infected person and a susceptible person that happens throughout the simulation. The dots on this piece are actual results from a singular run of my simulation.

**Aung Zaw Myint** (Myanmar)

**“Predator-prey Model ****ရဲ့**** အရေအတ့က်** **ပေါက်ကဲ့ခြင်း** **စနစ်**** “**(Burmese)

“Population Dynamics of Predator-prey System”

2019

digital

This Pheasant Math-Art is referred to as the predator-prey relationship system graphs of the nonlinear of differential equations in mathematical biology or ecology.

In the back part of Pheasant, the presented result graph between the lynx and the snowshoe hare the predator prey relationship. The yellow shows the population of lynx, while the red shows the population of hares. At the start of the graph, the lynx population was very high, which the hare population was relatively low (i.e. the snowshoe hare forms a large staple in the lynx diet. Without the hare, the lynx would starve. However, as the lynx eats the hare, or many hares, it can reproduce. Thus, the lynx population expands. With more lynx hunting, the hare population rapidly declines. )

For Lotka-Volterra equations, the Pheasant tails are referred to as a critical point in the first quadrant could be an asymptotically stable node or spiral point when the limiting population of the prey species in the absence of the predator species.

The front part of Pheasant referred to Prey-Predator dynamics as described by the level curves of a conserved quantity for Lotka-Volterra equations.

In the neck of Pheasant said that Kermack-McKendrick model of propagation of infectious disease. A short calculation shows that x(t) converges to a constant, say x(t)→x∗ where x∗ can be found by solving the equation C=rx∗-dlnx∗.

**Danilo Gregorin Afonso** (Brazil)

**“Resolvendo sem resolver”**(Portuguese)

“Solving without solving”

2019

Jigsaw puzzle

The idea behind this is the concept of a weak solution for elliptic partial differential equations. Since they are very difficult to actually solve, we change the concept of solution to a broader one, and then seek for theorems on existence and uniqueness of solutions. Later we study the regularity of these weak solutions. In some conditions, they will be actual solutions. In this way, we solve the problem without actually solving it. I hope the work to motivate discussions on this subject.

**Dragana Radojicic** (Serbia)

**“****As Einstein once said: “You can’t blame gravity for falling in love”, you shouldn’t blame mathematics for not doing well on stock markets****.”**2019

Pen on paper

As the sun goes up, we wake up wanting to know in which direction in the forest we should go in order to harvest some dollars from the tree. Should I plant, pick or just lay on the grass to get some tan? Should I buy, sell or idle? The stock market and high-frequency trading have attracted attention from the financial institutions and hedge funds, but also from academics. Before the sun goes down, we should decide which signs and indicators we take to trust them and believe in them.

The text in the picture is from the song: “Kralj izgubljenih stvari” in the performance of Serbian singer Sasa Matic, and it means: “you need a thousand steps to the top, and just one and only one to the bottom”.

**Jimoh Abdulganiyu** (Nigeria)

**“Eyi kii ṣe ọlọje x-ray awọn pixels.”**(Yoruba)

“This is not x-ray scan, it’s pixels.”

2019

drawing

During data collection in my research work I noticed that at every digital image taking section in radiography unit in hospitals, radiologist doctors have the strong belief that they are truly diagnosing patients for every image scan to detect illness but the doctors aren’t realising that what they think they are diagnosing are series of pixels representing or depicting the real medical image of patients. Therefore my artwork read “Eyi kii ṣe ọlọje x-ray awọn pixels.” & “This is not x-ray scan it’s pixels.”

**Lateef Olakunle Jolaoso** (Nigeria)

**“This is not a fixed-point theorem.”**2019

photography

My research focuses on the study of optimization and fixed point problems. Basically, designing/constructing iterative algorithms for the approximation of solutions of optimization and fixed point problems.

With this one can solve convex feasibility problems, split feasibility problems, variational inequalities problems, convex minimization problems, equilibrium problems, split monotone inclusion problems and split equality fixed point problems in Hilbert and Banach spaces. It lies between pure and applied mathematics.

**Matheus Pires Cardoso** (Brazil)

**“A caixa de Cantor”**(Portuguese)

“Cantor’s Box”

2019

digital

It is a simple box that looks like it is floating on a blue surface. This artwork tries to bring a simple explanation about one of the most incredible discoveries from set theory by Georg Cantor. The box could represent the set of positive integers, and inside of it all the subsets, the even numbers, Fibonacci’s numbers, odd numbers etc. All of them can be put in the box, and if we can put all those sets the box will be full but despite that happen you will be able to put the rationals and the negative numbers as well, even if the box is full. This can be explained with the concept of cardinality, the sets which we put in the box, all of them have the same cardinality and despite that, if we union two sets with the same cardinality the set that formed with the union of the two will have the same cardinality. Then we can put in more sets as long as they have the same cardinality of the other sets which already are in the box. That is the reason for the phrase “Mesmo cheia mas há espaço” in Portuguese, which can be translated to “Even if it is full, there is space for more”.

**Oshinubi Kayode** (Nigeria)

**“Kiraki”**(Yoruba)

“Crack”

2019

drawing

The artwork is all about cracks in structures as we know that cracks lead to the collapse of structures. The idea is to proffer solutions to cracks in structures in order to predict the location, size and depth of a crack.

**Ricardo Jose Sandoval Mendoza** (El Salvador)

**“Movimiento por Números”**(Spanish)

“Movement by Numbers”

2019

felt pen on paperboard

The primary focus of my research is data analytics on micro-mobility. In this artwork, I was thus trying to depict what has been the primary source of data for my research (dockless electric scooters) and what my research is about (data analytics). Even though there are many more components and tools that go into my research, I wanted my artwork to be pretty self-explanatory and easy for the viewer to get an idea of what I am doing at my lab. I chose to draw a dockless electric scooter, not only because it represents the data source for my research, but because dockless electric scooters are at the forefront of the micro-mobility revolution. They are changing the way in which people move around cities, and they are becoming a prominent mode of last mile transportation. It is thus important that we understand how people use them and the ways in which they interact with the built environment.

Der Beitrag Intercultural Science-Art Project – The Artworks erschien zuerst auf Heidelberg Laureate Forum.

]]>