24 Game Strategies, Possible and Impossible Cards, Factoring, Factorials, and Fractions

The 24 Game is an arithmetical card game in which the objective is to find a way to manipulate four integers so that the end result is 24. For example, for the card with the numbers 4, 7, 8, 8, a possible solution is (7-(8/8))x4=24 where I was looking for 6x4, but if you want to generate 8x3, then first 8-7=1 then 4-1=3, then 3x8=24.

The game has been played in Shanghai since the 1960s, using playing cards.

Additional operations, such as exponents, square root and factorial, allow more possible solutions to the game. For instance, a set of 1,1,1,1 would be impossible to solve with only the five basic operations. However, with the use of factorials, it is possible to get 24 as 1+1+1+1=4, then 4! Or 4 factorial is 4x3x2x1=24.

The official game rules are in the pdf as well as playing cards with one, two, and three dots. The dots represent the level of difficulty and you get the number of points associated with the level of difficulty.

I thought it would be interesting to have the children create their own 24 cards by first choosing a math fact such as 0+24, then create two cards that would make zero (8-8) and then two cards that would make 24 such as (6x4), then 0+24=24. Even though the traditional game only allows the digits 1-9 to make the four numbers, I allowed children to use larger numbers to create their cards.

I enjoyed watching the younger Mathletes look for patterns in the sums of:

0 + 24

1 + 23

2 + 22

And so on. Then I asked them why did I stop at 12 + 12 and many of them saw the Commutative Property at work, essentially that 11 + 13 is the same as 13 + 11. School teach these as turn around facts; it is better for them to learn the names of the properties early so they think mathematically. We also looked at the Commutative Property for multiplication and why it does not work for subtraction and division unless the two numbers are equal.

It was even more fun to look at the patterns of subtracting numbers starting from:

24 - 0

25 - 1

26 - 2

And so on. They were able to see that all they have to do is add the same number to 24 and to 0 to get the same difference of 24. For example 124 - 100 = 24.

I was very intrigued about why the game is called 24 and not some other number so I worked with the older Mathletes to figure out how many possible combinations of cards there are with the digits 1-9 without repeated combinations (for example, the cards 2,3,5,8 is the same no matter what order the numbers are in such as 5,2,8,3). If we didn’t care about the overlap, we could just multiply 9x9x9x9=9^4=6,581 cards; but we cannot ignore the overlap. So we looked at the number of separators between 9 digits (there are 8), added that to the four cards (8+4=12) the looked at the formula for 12 items taken 4 at a time with the notation, 12C4=12!/(8!4!)=495. The best part of this exploration was showing the Mathletes how to cancel like factors in fractions (8/8=1). So, we can essentially cancel 8!/8! Leaving us with 12x11x10x9/4! And then cancel 4x3=12 and simplify 10/2=5 leaving us with only 11x5x9 to multiply=495.

Then we looked at the number of cards that are considered impossible with the standard rules of just the four binary operations (+-x/); there are 91. So there are 404/495 cards solvable. I challenged the children to estimate the percentage of solvable cards so 400/500=4/5=80%. So the children made a conjecture that since so many cards are solvable that is why they chose 24. However, I found an analysis of target numbers from 0-99 with the same rules (someone created this spreadsheet in java script), and it showed 18 other target numbers with a higher percentage of solvable cards. For example, the number 2 has 99% solvable cards. But 97, a high prime number has only 8% of solvable cards.

I added a list of the 91 impossible cards on the last page of the pdf.

The beauty of this lesson was watching the Mathletes work their brains to solve these challenges. Most would start combining numbers to see what happened and usually would end up with solutions that were a few away from 24. Then I challenged them to think strategically like a chess player, and first identifying a math fact like 12x2 to solve. Then look for cards to make 2 and 12. This is so much more effective and hopefully that is the approach the Mathletes will use going forward. 

For the older Mathletes, one of the most beneficial outcomes of this lesson was introducing my algorithm for factoring numbers which I will continue in future lessons. I introduced this algorithm by asking them to first choose the factor pair that comes from the identity property of one (a x 1 = a).  So for 24:

1 x 24

2 x 12

4 x 6

8 x 3

So double the left column and take half of the number on the right column. This will always yield a factor pair of 24. We can continue this process, which we did with the older Mathletes to 16 x 3/2, then 32 x 3/4 indefinitely. However, non-whole numbers are not considered factors.

The difficult part is to have children unlearn the guess and check process they are taught in school so they start with 1, then 2, then 3, then 4, …. When children do this they invariably forget certain factors along the way. Try doing guess and check with the number 96, and then use my algorithm. You will see that my algorithm cannot miss a factor.

1 x 96

2 x 48

4 x 24

8 x 12

16 x 6

32 x 3

More to follow on my factoring algorithm when you hit an odd number on the right column.

Finally, for the older Mathletes, I showed them a few cards that seemed impossible but if you use fractions they are solvable. For example, the card: 3 8 3 8 has no solutions without fractions. First I subtracted 8/3 from 3 to get 1/3, then divided 8 by 1/3 which is the same as 8 x 3. This will be the beginning of students reinforcing how to divide fractions (multiply the dividend by the reciprocal of the divisor).

Another one is 3 3 7 7 where I first added 3 to 3/7 to get 24/7 and then multiplied that by 7.