# Thread: A Game Similar to 24 - Is it always solvable?

1. ## A Game Similar to 24 - Is it always solvable?

Hello,

I have recently learned a game thru a tutoring training class & am trying to figure out if all permutations are solvable? The game is using number only playing cards (2-9), set out 4 cards, plus 1 answer card. Object is to combine the first 4 cards ( add, subtract, multiply, divide) to equal the 5th answer card. Basically like the 24 game in the forum. My question is are all possible combinations solvable? My gut is yes, but curious how to prove it.

If it can't be proven, could anyone think of a way to write a program to try all the combinations? I thought of writting all the different math function combinations and then just substitute the 4 numbers 1 at a time, but even that seems like I'd be writing a whole lot of unique equations.

I apologize if this is an old idea, but not really sure how to event search for it without even a name of the game.

Andy

2. ## Re: A Game Similar to 24 - Is it always solvable?

I don't think they are all solvable.
Here are a few counter examples:
7 7 7 7 -> 7 (use four 7's to make 7)
9 9 9 9 -> 9
5 5 5 5 -> 9

In general you have to do a brute force search for all cases. but I am sure there are lots of unsolvable combinations

3. ## Re: A Game Similar to 24 - Is it always solvable?

Originally Posted by visionc
I don't think they are all solvable.
Here are a few counter examples:
7 7 7 7 -> 7 (use four 7's to make 7)
9 9 9 9 -> 9
5 5 5 5 -> 9

In general you have to do a brute force search for all cases. but I am sure there are lots of unsolvable combinations
A deck of playing cards has only 4 of each number card. So, the five cards cannot all be the same (unless the starting deck is a pinochle deck).

5+5-(5/5) = 9, so that example doesn't work, either.

2,2,2,2 cannot make 9.

4. ## Re: A Game Similar to 24 - Is it always solvable?

Originally Posted by SlipEternal
A deck of playing cards has only 4 of each number card. So, the five cards cannot all be the same (unless the starting deck is a pinochle deck).

5+5-(5/5) = 9, so that example doesn't work, either.

2,2,2,2 cannot make 9.
haha, good catch. but you get the drift.

7 7 7 7 -> 9
9 9 9 9 -> 7

neither can be solved. And lots of more such cases if one tries harder.

5. ## Re: A Game Similar to 24 - Is it always solvable?

On this site Facts about 24 the math game - 24theory, solves 24 the math game

There is a bit discussion why we play 24. It's not the most likely number to be solved. I guess it's the most fun number.

6. ## Re: A Game Similar to 24 - Is it always solvable?

I did a quick calculation. There are 2,632 different four-card draw plus answer card draws. Among them, 2,479 are solvable. The probability of drawing four cards and an answer card in a random draw from among 32 poker cards (only 2 - 9) is 96.4%. If someone would like to see the calculations, I made a simple Excel spreadsheet.

7. ## Re: A Game Similar to 24 - Is it always solvable?

SlipEternal,

I would be very interested in seeing how you did these calks. Thanks for the reply.

Andy