[Badcase]: Twenty-four game, how to get 24 from 23, 3, 11, 16 Not possible with the given constraints

[novohool]

Model Series

Qwen2.5

What are the models used?

QwQ-32B-preview

What is the scenario where the problem happened?

QwQ-32B-preview Final Answer: Not possible with the given constraints

Is this badcase known and can it be solved using avaiable techniques?

• I have followed the GitHub README.
• I have checked the Qwen documentation and cannot find a solution there.
• I have checked the documentation of the related framework and cannot find useful information.
• I have searched the issues and there is not a similar one.

Information about environment

online：
https://huggingface.co/spaces/Qwen/QwQ-32B-preview

Description

Testing URL

https://huggingface.co/spaces/Qwen/QwQ-32B-preview

user:

Twenty-four game, how to get 24 from 23, 3, 11, 16 by simple addition, subtraction, multiplication and division, using each number once

response:

So I've got this problem here: I need to make 24 using the numbers 23, 3, 11, and 16. I can only use each number once, and I can add, subtract, multiply, or divide them. Okay, let's see.

First, I should probably look at the biggest number, which is 23, and see how it relates to 24. It's just one less than 24, so maybe if I add 1 to it, I get 24. But I don't have a 1 in my numbers. Hmm.

Wait, maybe I can make a 1 using the other numbers. For example, if I subtract 3 minus 3, that's 0, but that won't help. Or 11 minus 11 is 0. Not useful. Maybe division? Like 3 divided by 3 is 1, but again, that's using two threes, which I don't have. So that's not helpful.

Let me think differently. Maybe I can use multiplication to get close to 24 and then adjust with addition or subtraction.

For example, 11 times 3 is 33. That's higher than 24. Maybe I can subtract something from it. Subtracting 23 would give me 33 - 23 = 10, which isn't 24. Subtracting 16 would be 33 - 16 = 17, still not 24.

Alternatively, 11 times 2 is 22, which is close to 24. But I don't have a 2, I have 23 and 16 left. Wait, maybe I can use 23 minus something to get to 24.

Wait, that doesn't make sense. If I have 23 and I add something to it, I can get to 24 by adding 1, but I don't have a 1. Or maybe I can get a fraction that adds up appropriately.

Maybe I should look at division. For example, 23 divided by something and then added to or subtracted from another number.

Wait, 23 divided by 3 is approximately 7.666, which doesn't seem helpful. Maybe I can divide 16 by something.

Or perhaps I can consider multiplying 3 and 11 to get 33, then do something with 23 and 16.

Let me try adding or subtracting 23 and 16. 23 + 16 is 39, which is higher than 24. 23 - 16 is 7, which is lower.

Maybe I can combine these in some way with the other numbers.

Wait, perhaps if I take 33 (from 3 times 11) and subtract 23 - 16 is 7, so 33 - 7 is 26, which is close to 24, but not quite there.

Alternatively, maybe I can consider dividing 23 by 16. 23 divided by 16 is 1.4375. Not helpful.

Wait, maybe I can consider multiplying 3 and 11 to get 33, then subtract 23 to get 10, and then see what to do with 16.

But 10 plus 16 is 26, which is too high. 10 minus 16 is -6, too low. Not helpful.

Let me try another approach. Maybe I can use 16 and 11 to get something useful.

16 plus 11 is 27, which is close to 24. If I can subtract something to get to 24, but I only have 23 and 3 left. 27 minus 3 is 24. That's interesting.

So, 16 + 11 = 27, minus 3 = 24. But I still have the 23 left, which isn't used in this case.

Wait, the problem says to use each number once, so I need to incorporate all four numbers: 23, 3, 11, and 16.

So, my previous step only uses three numbers, which isn't allowed.

Hmm, okay, back to the drawing board.

Maybe I can consider using division with 23.

For example, 23 divided by something. Wait, 23 is a prime number, so its only divisors are 1 and itself. So dividing by 3 would give a decimal, which might not be helpful.

Alternatively, maybe I can make a fraction that cancels out appropriately.

Wait, perhaps if I consider 23 minus something divided by something.

Let me try that.

For example, 23 minus (11 divided by something). But I don't have another number to divide 11 by.

Wait, maybe 23 minus (11 divided by 3), but 11 divided by 3 is approximately 3.666, and 23 minus that is about 19.333, which isn't helpful.

Alternatively, maybe I can divide 16 by 3, which is approximately 5.333, and then do something with that.

But again, that doesn't seem to lead me to 24.

Maybe I should look for a way to combine the numbers differently.

Let me try multiplying 3 and 16, which is 48, then see what to do with 23 and 11.

48 minus 23 is 25, which is close to 24, but not quite there.

Alternatively, 48 minus 11 is 37, which is too high.

Wait, maybe I can divide 48 by something to get closer to 24.

24 is half of 48, so maybe if I can divide 48 by 2, but I don't have a 2.

Wait, perhaps I can make a 2 using the other numbers.

For example, 23 minus 11 is 12, and 12 divided by 6 is 2, but I don't have a 6.

Alternatively, 23 minus 16 is 7, which isn't helpful.

Hmm.

Maybe I can consider 23 plus something divided by something.

For example, (23 + 11) divided by something. 23 + 11 is 34. Dividing 34 by something to get 24 seems unlikely, as 34 divided by something would need to be less than 1, which would require a number larger than 34, which I don't have.

Wait, 34 divided by something to get 24? That would mean something like 34 / x = 24, so x = 34 / 24, which is approximately 1.4167. But I don't have a number that divides into 34 to get that.

This seems too convoluted.

Let me try another approach.

Maybe I can use 16 and 3 to get something useful.

16 times 3 is 48, which is twice 24. Maybe I can divide that by something.

But again, I don't have a number that divides into 48 to get 24 directly.

Wait, 48 divided by 2 is 24, but I don't have a 2.

Alternatively, maybe I can make a 2 using the other numbers.

For example, (23 + 11) divided by 17 is approximately 2, but I don't have a 17.

Wait, that's not helpful.

Alternatively, maybe I can consider 23 minus 11, which is 12, and 12 divided by 6 is 2, but again, no 6.

This isn't working.

Maybe I should look at addition and subtraction first.

For example, 23 + 3 is 26, then add 11 to get 37, then subtract 16 to get 21. Not 24.

Alternatively, 23 + 11 is 34, minus 16 is 18, plus 3 is 21. Still not 24.

Wait, maybe another order.

23 minus 11 is 12, plus 16 is 28, minus 3 is 25. Close, but not 24.

Alternatively, 23 minus 16 is 7, plus 11 is 18, plus 3 is 21. Again, not 24.

Hmm.

Maybe I need to use division in a different way.

For example, 23 minus (16 divided by something). 16 divided by 3 is approximately 5.333, and 23 minus that is about 17.666, which isn't helpful.

Alternatively, 16 divided by 11 is approximately 1.4545, and 23 minus that is about 21.545, not useful.

Maybe I can consider multiplying 11 and 3 to get 33, then subtracting something.

33 minus 23 is 10, plus 16 is 26, minus 3 is 23. Still not 24.

Wait, maybe if I multiply 11 and 2, I get 22, which is close to 24, but I don't have a 2.

Alternatively, maybe I can consider 16 plus 8 is 24, but I don't have an 8.

Wait, maybe I can make an 8 using the other numbers.

For example, 23 minus 15 is 8, but I don't have a 15.

Alternatively, 23 minus 11 is 12, and 12 divided by something to get 8? That would require dividing by 1.5, which I don't have.

This is getting tricky.

Maybe I need to think about factors of 24.

24 can be factored into 4 times 6, or 3 times 8, or 2 times 12.

But I don't have those numbers directly.

Wait, maybe I can make 4 and 6 using the given numbers.

For example, 16 divided by 4 is 4, but I don't have a 4.

Alternatively, 23 minus 19 is 4, but again, no 19.

Hmm.

Alternatively, maybe I can make 3 and 8.

For example, 24 is 3 times 8.

I have a 3, but how can I make an 8 using 23, 11, and 16?

Wait, 16 divided by 2 is 8, but I don't have a 2.

Alternatively, 11 plus 3 is 14, which isn't helpful.

Wait, maybe 23 minus 11 is 12, and 12 minus 4 is 8, but again, no 4.

This is frustrating.

Maybe I should consider more complex operations, like adding and subtracting in a specific order.

For example, start with 23, add 11 to get 34, subtract 16 to get 18, then add 3 to get 21. Not 24.

Alternatively, 23 plus 3 is 26, minus 11 is 15, plus 16 is 31. Still not 24.

Wait, maybe another order.

Take 3 times 11, which is 33, minus 23 is 10, plus 16 is 26, minus 3 is 23. No good.

Alternatively, 33 minus 16 is 17, plus 23 is 40, minus 11 is 29. Not helpful.

This seems really tough.

Maybe I need to use division in a different way.

For example, (23 - 11) divided by something.

23 minus 11 is 12. 12 divided by 3 is 4, and then maybe do something with 16.

4 times 16 is 64, which is way above 24.

Alternatively, 12 plus 16 is 28, minus 3 is 25, not 24.

Wait, maybe if I can get to 24 using three numbers and then incorporate the fourth number in a way that doesn't change the result.

But that seems unlikely.

Alternatively, maybe I can use subtraction in a creative way.

For example, 23 minus (something minus something).

But I'm not sure.

Wait, maybe if I take 23 and add something, then subtract something else.

For example, 23 plus 16 is 39, minus 11 is 28, minus 3 is 25. Still not 24.

Alternatively, 23 plus 11 is 34, minus 16 is 18, plus 3 is 21. No good.

Wait, maybe if I multiply 3 and 16 to get 48, then subtract 23 to get 25, then subtract 11 to get 14. Not helpful.

Alternatively, 48 minus 24 is 24, but I need to get to 24 using all four numbers.

Wait, maybe if I can make 24 using three numbers and then show that the fourth number doesn't affect it, but that seems impossible.

Alternatively, maybe I can use division to scale things appropriately.

For example, if I have 48 divided by 2 is 24, but again, I don't have a 2.

Wait, maybe I can make a 2 using 11 and 3.

For example, 11 minus 3 is 8, and 8 divided by 4 is 2, but I don't have a 4.

Alternatively, 11 minus 9 is 2, but I don't have a 9.

This is getting too complicated.

Maybe I need to consider that it's not possible with these numbers.

Alternatively, maybe I'm missing a simple combination.

Let me try adding all the numbers together: 23 + 3 + 11 + 16 = 53, which is way above 24.

Alternatively, multiplying 3 and 11 to get 33, then subtracting 16 to get 17, then adding 23 to get 40, which is still not 24.

Wait, maybe if I take 33 minus 23 is 10, then add 16 to get 26, minus 3 is 23. Still not 24.

This is driving me crazy.

Maybe I need to accept that it's not possible with these numbers using simple operations.

Alternatively, perhaps there's a more creative way involving multiple operations in a specific order.

For example, (23 - 11) * (16 / 3). That's 12 * (16 / 3) = 12 * 5.333..., which is 64, not 24.

Alternatively, (23 + 11) - (16 / 3). That's 34 - 5.333... = 28.666..., not 24.

Wait, maybe if I take 23 minus (16 / 3), which is 23 - 5.333... = 17.666..., then add 11 to get 28.666..., still not 24.

This isn't working.

Maybe I need to use division in the numerator or denominator in a fraction.

For example, (23 - 11) / (16 / 3) = 12 / (16 / 3) = 12 * (3 / 16) = 36 / 16 = 2.25, which is not 24.

Alternatively, (23 + 11) / (16 / 3) = 34 / (16 / 3) = 34 * (3 / 16) = 102 / 16 = 6.375, still not 24.

Hmm.

Maybe I need to consider that it's not possible with these numbers using the allowed operations.

Alternatively, perhaps there's a mistake in my calculations, and I need to try again.

Wait, maybe if I take 23 plus something and then divide.

For example, 23 + 16 is 39, divided by something.

If I divide 39 by something to get 24, that would require 39 / x = 24, so x = 39 / 24, which is 1.625.

But I don't have a 1.625, and I need to use the other numbers.

Alternatively, maybe I can make 1.625 using 11 and 3.

For example, 11 divided by 3 is approximately 3.666, which isn't helpful.

Alternatively, 3 divided by 11 is approximately 0.2727, which is too low.

This isn't working.

Maybe I need to consider that it's not possible and look for an alternative approach.

Alternatively, perhaps I can use subtraction in a different way.

For example, 23 minus (11 - something).

Wait, 23 minus (11 - 3) = 23 - 8 = 15, which isn't 24.

Alternatively, 23 minus (11 / 3), which is approximately 23 - 3.666 = 19.333, not helpful.

Wait, maybe if I take 23 minus (16 / something).

For example, 16 divided by 3 is approximately 5.333, and 23 minus that is about 17.666, which isn't useful.

Alternatively, maybe I can consider multiplying 23 and something and then adjusting with another number.

For example, 23 times 3 is 69, then subtract something.

69 minus 11 is 58, minus 16 is 42, not 24.

Alternatively, 69 minus 16 is 53, minus 11 is 42, still not 24.

This is really challenging.

Maybe I need to think outside the box.

For example, perhaps using division to create a fraction that can be multiplied or added to get 24.

Wait, for instance, if I take 16 divided by 3, which is approximately 5.333, and then do something with that.

But I don't see a clear path to 24.

Alternatively, maybe I can consider 23 as 24 minus 1, and see if I can make 1 using the other numbers.

But as I thought earlier, I don't have a way to make 1 from 3, 11, and 16.

Wait, maybe if I take 11 minus (16 / 3). 16 divided by 3 is approximately 5.333, and 11 minus that is about 5.666, which isn't helpful.

Alternatively, maybe I can consider 16 minus (11 / 3). 11 divided by 3 is approximately 3.666, so 16 - 3.666 is about 12.333, which isn't useful.

This is really tough.

Maybe I need to accept that it's not possible with these numbers using simple addition, subtraction, multiplication, and division.

Alternatively, perhaps there's a mistake in my approach, and I need to try a different combination.

Wait, maybe if I take 23 and add something, then divide.

For example, 23 plus 11 is 34, divided by something.

If I divide 34 by something to get 24, that would require 34 / x = 24, so x = 34 / 24, which is approximately 1.4167.

But I don't have a way to make 1.4167 from 3 and 16.

Alternatively, maybe I can make it by multiplying or dividing those numbers.

Wait, 16 divided by 3 is approximately 5.333, which is not helpful.

Alternatively, 3 divided by 16 is 0.1875, which is too small.

This isn't working.

Maybe I need to consider that it's not possible and look for an alternative approach.

Alternatively, perhaps I can use subtraction in a different way.

For example, (23 - 11) is 12, then (12 * something) to get 24.

12 times 2 is 24, but I don't have a 2.

Alternatively, 12 times (16 / something), but I don't see a way to make that work.

Wait, 12 times (16 / 3) is 12 * 5.333..., which is 64, not 24.

Alternatively, 12 times (3 / 16) is 12 * 0.1875 = 2.25, which isn't helpful.

This is really tricky.

Maybe I need to consider that it's not possible with these numbers using the allowed operations.

Alternatively, perhaps there's a more creative way involving multiple steps that I haven't thought of yet.

For example, maybe I can use the result of one operation as part of another operation in a more complex expression.

Wait, for instance, (23 - (11 / 3)) * (16 / something). But I don't see how that helps.

23 minus (11 / 3) is approximately 23 - 3.666 = 19.333, and then multiplying by (16 / something) doesn't seem promising.

Alternatively, maybe I can consider adding and subtracting in a specific order to cancel things out.

For example, 23 plus 11 is 34, minus 16 is 18, plus 3 is 21. Still not 24.

Alternatively, 23 plus 3 is 26, minus 11 is 15, plus 16 is 31. No good.

Wait, maybe if I take 23 minus 16, which is 7, then add 11 to get 18, then add 3 to get 21. Still not 24.

This is really frustrating.

Maybe I need to accept that it's not possible and look for an alternative approach.

Alternatively, perhaps there's a mistake in my calculations, and I need to try again.

Wait, maybe if I take 23 plus something divided by something else.

For example, (23 + 11) / (16 / 3) = 34 / (16 / 3) = 34 * (3 / 16) = 102 / 16 = 6.375, which isn't 24.

Alternatively, (23 + 3) / (16 / 11). 23 + 3 is 26, and 16 / 11 is approximately 1.4545, so 26 / 1.4545 is approximately 17.857, not 24.

This isn't working.

Maybe I need to consider that it's not possible with these numbers using the allowed operations.

Alternatively, perhaps there's a more advanced mathematical operation that could be used, but since the problem specifies simple addition, subtraction, multiplication, and division, I should stick to those.

Given that, I think it's not possible to reach 24 using the numbers 23, 3, 11, and 16 with each number used once and the allowed operations.

**Final Answer**

\[ \boxed{\text{Not possible with the given constraints}} \]