A Puzzle ‘Four’ the New Year
It’s a New Year, and with it comes a new four-digit number. When faced with a number like 2019, it’s the mathematician’s natural instinct to do maths with it. Having quickly checked whether the year is prime (it’s sadly divisible by 3) the next obvious step is to discover interesting facts about the number, and to create and share number puzzles which use it.
Alex Bellos posted a set of number puzzles on New Year’s Eve, including Ed Southall’s lovely fact that 2019 is the smallest number that can be written in 6 ways as the sum of the squares of 3 primes; and Matt Parker’s posted a YouTube video listing some interesting number facts about 2019, in 2 minutes and 19 seconds.
One of Alex Bellos’ puzzles is a real classic of the genre – I see the same puzzle popping up every year, each time using the digits of the year in question:
Using only the digits 2, 0, 1 and 9 [exactly once each], create expressions that equal all of the numbers from 0 to 12. The expressions can include any of the arithmetical symbols +, –, x, ÷ and √, and brackets.
I’ll start you off:
- 0 x (2 + 1 + 9) = 0
- 2 – 1 + (0 x 9) = 1
If you’d like to spend some time working on this problem, go ahead – you can check your answers against Alex’s solution post. This type of problem – “Using only the given digits and certain operations, which numbers can be made?” – will be familiar to viewers of Channel 4’s Countdown in the UK as the regular Numbers Game.
It’s a Numbers Game
In the show, a mathematician places a selection of randomly chosen numbers across the top of the board, and then presses a button to generate a random number as a target. Contestants have 30 seconds to work out how to combine them, using addition, subtraction, multiplication and division only, to reach the target.
It’s a lovely challenge, and still hugely popular after many years of broadcast. For a nice example of an out-of-left-field solution, watch this excellent example from 1997, and at the other end of the spectrum, watch this clip from 2009 in which the random number generator lands on a particularly easy one (featuring a beautiful 30 seconds of British people sitting around in awkward silence, as we do so well).
The rules of the Countdown Numbers Game have been well formulated – it only allows the use of the four basic mathematical operations, and the numbers are chosen from a predetermined set (the ‘top row’ referred to in the clip always includes 25, 50, 75 and 100 and the rest of the board – ‘small ones’ – consists of the numbers 1-10, twice each). Contestants can choose any configuration of six numbers they like, picked randomly from the large and small (e.g. ‘two large and four small’).
Changing the Rules
Such strict rules work well in a competition setting, but in our 2019 puzzle context, it wouldn’t be as interesting. Alex’s puzzle specifies that your expressions for the numbers 1-12 “can include any of the arithmetical symbols +, –, ×, ÷ and √, and brackets” – he’s included the square root symbol here, since it doesn’t involve writing any numbers, so it’s not as much like cheating as it would be if you included the ‘squared’ symbol as well.
In order to reach the numbers 13-20, Alex allows a little more leeway (presumably because some of the numbers in this range aren’t possible with the initial set of operations) – you’re now allowed to concatenate numbers together (for example, you can put the digits 1 and 9 together to make 19) and put numbers as exponents – so you would be allowed to square something, since 2 is one of your digits (but you couldn’t raise something to a power you don’t have a digit for).
Alex then invites you to take this even further, and add in other mathematical expressions to get all the numbers up to 100. There’s a project for a rainy day!
‘Four’ Another Challenge
The genre of puzzles this 2, 0, 1, 9 digit puzzle falls in has its own classic version – originally printed in “Mathematical Recreations and Essays” by W. W. Rouse Ball back in 1914, the Four Fours puzzle challenges people to make each whole number using four of the number 4. Without giving too much away, the first few examples can be calculated as follows:
0 = 4 ÷ 4 × 4 − 4
1 = 4 ÷ 4 + 4 − 4
2 = 4 − (4 + 4) ÷ 4
3 = (4 × 4 − 4) ÷ 4
Of course, each of these can be expressed in several other ways, and all the numbers up to 72 are definitely possible, given the right set of operations – and many others are possible above that.
Operations you might permit include:
- The standard ‘Countdown’ mathematical operations of +, – , × and ÷
- Concatenation of digits to make two, three and four digit numbers
- Exponentiation – putting one number to the power of another
- Taking the square root, or the nth root, if n is a digit you’re allowed
- Factorial symbol, e.g. 4! = 4 × 3 × 2 × 1
- Decimal points, to produce e.g. .4 (point four, with an implied nought) = 4/10
- A dot or bar over a number to indicate an infinite recurring decimal, or a … at the end, so you can make .4… = 4/9
Some people also include some slightly questionable options – for example, the subfactorial, !n, also called the derangement number, is defined as the number of ways to rearrange a set of n objects so that none of them end up in their original places – for example, 1234 going to 2341. !4 = 9, and while this is a well-defined quantity, it’s rarely a function that’s seen or used outside of serious pure mathematics, and very few people have heard of it.
Similarly, the gamma function, Γ(x), is considered an extension of the factorial to non-integer values – for example, you could calculate Γ(0.4) = 2.21816… but it wouldn’t be much use in trying to make 5. However, the gamma function is still defined on whole numbers – but because of the way it’s defined, Γ(n) = (n-1)!. This means Γ(4) = 6, which might get you out of a tricky spot if you need a six and haven’t got enough fours left to make it. But it’s surely cheating!
Once you’ve decided which of these to allow and how constrained to make the challenge, you can attempt Four Fours (or its more challenging alternatives, Five Fives and Six Sixes, which I hope I don’t have to explain); or you could pick any four digits – your birth year might make for a nice personalised puzzle.
Some coders have taken it upon themselves to find ways to crunch the problem through software – given your permitted operations and starting digits, you can apply all the allowed functions to all the numbers (or pairs of numbers) you start with, then repeat this using your results – having done this enough times you’ll find all the possible numbers you can get using the given starting criteria. But that’s DEFINITELY cheating.
A Game of Four Fours
Given the idea of picking any four digits to play with, it’s also possible to turn this into a competitive game, to challenge your friends and see who’s the best at this. You wouldn’t be the first to do this – Krypto, a game designed by Daniel Yovichin 1963, involves dealing a set of cards in front of the players to pick your numbers, and players compete to find expressions for given sets of cards before their friends do.
If you’re not interested in buying a special set of cards to play, it’s also possible to play using a normal deck of playing cards. Number fighting fans in Shanghai developed The 24 Game (not to be confused with 24: The Game, which is the official board game of the Kiefer Sutherland TV series, and is very different) – using a standard deck of cards with pictures removed, four random cards are dealt onto the table in front of everyone, taking aces to be 1, and whoever can make 24 first is the winner.
24 has been chosen here as a number with plenty of factors, that can be reached in a variety of ways by multiplication and addition, and with a bit of quick thinking it’s possible to work out a way to get 24 from almost all of the 1820 four-card combinations (but not all: sadly, 1, 1, 1, 1 isn’t possible with the standard arithmetical operations – unless you allow factorials).
My friends and I have played a hand-based version, where each of four players is dealt 10 cards, and on each turn everyone places a single card into the middle – whoever shouts a way to make 24 first wins those four cards and scores them in a pile, and once all 10 cards from your hands have been played, the winner is whoever has the most scored cards. (By agreement, it’s occasionally been necessary to give up on a particular set of four numbers, when nobody has been able to get an answer – in which case you can score one each).
You might find the idea of this kind of number torture bemusing – mathematics is about much more than just mental arithmetic and crunching numbers, and since you can choose the rules you give yourself, the challenge is slightly arbitrary. Even so, blackboards in maths departments all over the world will find that if someone started writing “4 4 4 4 = 1, 4 4 4 4 = 2″ down one side of the board, leaving gaps to fill in the operators, it would quickly become a project with which many would join in.