# Very Hard Math Puzzle

• Jan 9th 2010, 08:26 AM
stones44
Very Hard Math Puzzle
So you have 6 numbers, ranging from 1-24. Five of the six numbers must equal the 6th number by using any of the 4 operations (+, -, *, /) and using each number only once.

So for example:

3 7 12 19 21 | 5

So you need to get the number 5 by moving around and using the operations on those five first numbers.

Now, find an example where this is impossible!

EDIT: I think I found one:

5 11 17 19 23 -> 1

See if you can find away this would work, cause I cannot!

Also, how would one go about proving it is a valid counter example?
• Jan 9th 2010, 11:41 AM
wonderboy1953
Suggestion
Shouldn't this go under the Math Challenge Problems section?
• Jan 9th 2010, 02:06 PM
pomp
Quote:

Originally Posted by stones44
so you have 6 numbers, ranging from 1-24. Five of the six numbers must equal the 6th number by using any of the 4 operations (+, -, *, /) and using each number only once.

So for example:

3 7 12 19 21 | 5

so you need to get the number 5 by moving around and using the operations on those five first numbers.

Now, find an example where this is impossible!

Spoiler:
1,5,9,17,23 | 24
• Jan 9th 2010, 03:02 PM
sym0110
Quote:

1,5,9,17,23 | 24
Surely (23-17)*(9-5)/1=24... you need to find something else, 24 has too many factors
• Jan 9th 2010, 03:24 PM
sym0110
consider 1 6 11 16 21 | 18

I've thought it over many times and think it works well as a counter example
• Jan 9th 2010, 03:50 PM
pomp
Quote:

Originally Posted by sym0110
Surely (23-17)*(9-5)/1=24... you need to find something else, 24 has too many factors

For some reason in my head I was thinking you can only add or subtract a 1, because multiplication or division by 1 were pointless (Headbang) Nevermind.

I don't think 24 having too many factors is a fair reason to rule it out though, if you pose the same problem but with 4 numbers instead of 5, then there are plenty of combinations of 4 numbers that can never be made to equal 24.
• Jan 9th 2010, 03:53 PM
pomp
Quote:

Originally Posted by sym0110
consider 1 6 11 16 21 | 18

I've thought it over many times and think it works well as a counter example

A counterexample to what?

If you mean a solution to the problem, you missed:

21 - (16 - 1) / (11 - 6) = 18
6 * ((21 + 11) / 16 + 1) = 18
11 * (6 - 1) - 21 - 16 = 18
21 - (16 + 6) / 11 - 1 = 18
6 * (21 - 16) - 11 - 1 = 18
16 + (21 - 11) / (6 - 1) = 18
• Jan 11th 2010, 06:03 PM
stones44
EDIT: I think I found one:

5 11 17 19 23 -> 1

See if you can find away this would work, cause I cannot!

Also, how would one go about proving it is a valid counter example?
• Jan 11th 2010, 06:10 PM
pomp
Quote:

Originally Posted by stones44
edit: I think i found one:

5 11 17 19 23 -> 1

see if you can find away this would work, cause i cannot!

also, how would one go about proving it is a valid counter example?

(17 + (19 + 11) / 5) / 23 = 1
17 / ((19 - 11) * 5 - 23) = 1
((23 - 17) * 5 - 11) / 19 = 1
(11 + (23 + 17) / 5) / 19 = 1
11 / ((23 - 17) * 5 - 19) = 1
(19 - 11) * 5 / (23 + 17) = 1
(23 - 17) * 5 / (19 + 11) = 1
(19 + 11) / (23 - 17) / 5 = 1
((19 - 11) * 5 - 23) / 17 = 1
(23 - (19 + 11) / 5) / 17 = 1
23 / ((19 - 11) * 5 - 17) = 1
11 / (19 - (23 + 17) / 5) = 1
19 / (11 + (23 + 17) / 5) = 1
(23 - 11) / (19 - 17) - 5 = 1
17 / (23 - (19 + 11) / 5) = 1
23 / (17 + (19 + 11) / 5) = 1
• Jan 11th 2010, 06:12 PM
stones44
how are you coming up with all these so quickly? do you have some method or program?

also how could one go about proving a counter example? (or proving there are none)
• Jan 11th 2010, 06:22 PM
pomp
Quote:

Originally Posted by stones44
how are you coming up with all these so quickly? do you have some method or program?

also how could one go about proving a counter example? (or proving there are none)

I have written a program that can check them.

As for proving a counter example or proving there are none, that's rather difficult I think.

The only way I can think of showing that a solution is true is by an exhaustive algorithm. If someone were to say they have a solution, I can verify it computationally using a program that exhausts all possible parsings of the numbers and symbols, and shows none of them equal the target number.

Whether or not you see this as a proof is your call, you'd have to have confidence that my program really did check all possibilities. (Wink)
• Jan 11th 2010, 06:56 PM
sym0110
Quote:

Originally Posted by pomp
I have written a program that can check them.

As for proving a counter example or proving there are none, that's rather difficult I think.

The only way I can think of showing that a solution is true is by an exhaustive algorithm. If someone were to say they have a solution, I can verify it computationally using a program that exhausts all possible parsings of the numbers and symbols, and shows none of them equal the target number.

Whether or not you see this as a proof is your call, you'd have to have confidence that my program really did check all possibilities. (Wink)

True there aren't many ways of organizing the +-/* and brackets with 4 numbers, and there are only 24C5=42504 ways to exhaust all possibilities. It's probably only going to take a few hours to check all of them with a not so efficient program. But I don't think computer provided exhaustion is really a proof...
• Jan 11th 2010, 07:02 PM
pomp
Quote:

Originally Posted by sym0110
True there aren't many ways of organizing the +-/* and brackets with 4 numbers, and there are only 24C5=42504 ways to exhaust all possibilities. It's probably only going to take a few hours to check all of them with a not so efficient program. But I don't think computer provided exhaustion is really a proof...

But we're using + - / * on 5 numbers, with a 6th as a target. There are so many combinations.

Why wouldn't you see this as proof? Do you think the 4 colour theorem proof is not really a proof? It's not the most elegant of proofs, actually, it's probably the most inelegant proof I've ever heard of, but it's proof nonetheless.