Quacky and Mario (a solution)

This is my solution to this problem, given here due to it's length.

Wow, this is such a clever puzzle! However, it's going to be really hard to explain the solution to, so apologies in advance. The more you write out yourself, the easier I think it'll be. For instance, I found that writing lists of numbers and scratching out a number every time they're eliminated really helped.

At the start of the problem, here's what Quacky knows:

  • Mario's sum is anywhere from 2\boldsymbol{2} to 16\boldsymbol{16}.

And here's what Mario knows:

  • Quacky's product is anywhere from 1\boldsymbol{1} to 64\boldsymbol{64}.

I'll organize this long solution into "rounds" (since we're eliminating things).

Round 1Round~1

Let's start with the first statement. Quacky says, "I don't know the numbers". Here's what we can now say about Quacky's product:

  • Quacky's product cannot be prime, since he would instantly know what the two numbers were (every prime's factors are just 11 and itself) for primes less than 9. Primes greater than 9 are also impossible, since one of the numbers will be greater than 88. So he can eliminate all the primes between 11 and 6464.
  • Quacky's product cannot be a number which can only be written as the product of two numbers one way. For example, if Quacky's number was 99, there would only be one product of two numbers, 3×33\times3, since the two numbers cannot be more than 88. Thus, Quacky would have said he knew what the two numbers were if his product was 99. He didn't however, so any numbers like this can be eliminated.

If we go through all of Quacky's potential products now and eliminate everything we can, we end up with this list of products: 4,6,8,12,16,24\boldsymbol{4, 6, 8, 12, 16, 24}. We've made good progress! :)

Here's what we can also say about Mario's sum:

  • If we take each of those five products we just came up with and list all the pairs of numbers they could be made from, we can then take each potential pair and add the two numbers to find a potential sum. For example, the number 66 could be made from 1×61\times6 or 2×32\times3. Adding each pair together, we see that Mario's sum could be either 77 or 55. If we do this for each potential product, we end up with the following relatively short list of potential sums: 4,5,6,7,8,9,10,11\boldsymbol{4, 5, 6, 7, 8, 9, 10, 11}.

Round 2Round~2

Next, Mario says, "I don't know the numbers". Here's what we can now say about his sums:

  • If we go through each potential sum we just found, we can list the pairs of numbers which could make up that sum and eliminate any with only one pair that works. For example, 99 could be 1+81+8, 2+72+7, 3+63+6, or 4+54+5. But if we try to take the product of each of those pairs, we can see that only one of them, 11 and 88, would result in a product already in our potential products list (the rest were already eliminated). We can eliminate 44, 99, and 1111 this way, leaving us with these potential sums: 5,6,7,8,10\boldsymbol{5, 6, 7, 8, 10}.

Round 3Round~3

Next, Quacky says, "I don't know the numbers" (for the second time). We can now say this about Quacky's products:

  • If we go through each potential product, we can list the pairs of numbers which could make up their product and eliminate any with only one pair that works, kind of like last time I guess :). For example, 88 could be 1×81\times8 or 2×42\times4. But the sum of 11 and 88 is 99, and we've already eliminated 99 from our potential sums. So there's only one pair of numbers which could make 88, but since Quacky just said he still doesn't know what the two numbers are, the product can't be 88. Otherwise, he would have said that he knew what the two numbers were. Doing this, we can eliminate 44, 88, and 2424, leaving this list of potential products: 6,12,16\boldsymbol{6, 12, 16}. Getting close! :)

Round 4Round~4

Next Mario says , "I don't know the numbers" (for the second time). We'll do the exact same process as Round 2 here, eliminating sums with only one possible pair of numbers that work (again, since Mario would know what the two numbers if that was the case). Doing this eliminates 55, 66, and 1010, which leaves only two potential sums: 7,8\boldsymbol{7, 8}.

Round 5Round~5

For the third and final time, Quacky says, "I don't know the numbers". This time we'll repeat the process of Round 3.

  • Doing so eliminates 66 and 1616, which means that we now know the product, 12\boxed{\boldsymbol{12}}!

Round 6Round~6

In a surprising turn of events, Mario says, "I know the numbers". How can he say this? Because he now has both Quacky's product and his own sum which he always knew. However, as @pratibha srivastava pointed out, we, the viewers, will never be able to figure out what exactly the two numbers were! We know that Mario's sum is either 77 or 88, but neither can be eliminated. Thus, the most we can say about the two numbers is that they are either 33 and 44 or 22 and 66.

And we're done! Like I said, it takes a really long time to explain, but the principles are relatively simple. This has to be one of my absolute favourite puzzles I've ever seen. On the surface, it looks just impossible! Hopefully this solution was clear enough. Feel free to ask for clarification.

Note by David Stiff
2 weeks, 4 days ago

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Thanks, David for the well-explained solution! In round 6, How did you eliminate 8, given that both 8(6+2) and 7(4+3) have only one pair of numbers that multiply to 12?

pratibha srivastava - 2 weeks, 3 days ago

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Oh, good catch! I assumed that we could repeat the Round 2 process one last time, but Mario said "I know the numbers", not "I don't know the numbers", so we can't eliminate one-pair sums this way (and I somehow overlooked 6+26+2 anyway).

I think the correct explanation for why Mario now knows the numbers is that he already knows his own sum! Given that and Quacky's product which he figured out, he will be able to figure out the two numbers.

But this means something interesting for us, the viewers...we'll never know the two numbers, or Mario's sum! Like you pointed out, both 77 and 88 are candidates, but we can't say which is his, and thus what the two numbers are. I guess that's why the puzzle asked for the product, and not the two numbers.

Thanks for pointing this out! I'll update the solution.

David Stiff - 2 weeks, 3 days ago

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Well thanks!!! (once again):)

pratibha srivastava - 2 weeks, 3 days ago

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@Pratibha Srivastava You're welcome! I changed one more thing, this time in Round 1 where we eliminate primes. I realized that my explanation was only for primes less than 99, but there is a similar explanation for primes greater than 99, so I added that. :)

David Stiff - 2 weeks, 3 days ago

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@David Stiff Well, thanks for updating the solution! !(your solutions are well specified and insightful ):)

pratibha srivastava - 2 weeks, 1 day ago

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