[Logic Puzzle] How old is Mom?

[Mike]

Heather and her son Marcus have just finished cleaning up after a holiday dinner.

Marcus says, "I've never met your cousin Cory before. How old is everyone in her family?"

Heather says, "Cory, Dave and Ashley? What if I told you the product of their ages is 2450? Would that help?"

Marcus does a few calculations and replies, "I still don't know."

Heather says, "Well, the sum of their ages is two times your age."

Her son thinks and says, "That still isn't enough information."

Heather then says, "Do you remember how I didn't eat any cake at my last birthday party, because I'm trying to be healthier? I wonder if Cory and her family will stop eating cake when they reach my age."

Marcus says, "OK... NOW I know their ages!"


How old is Heather?

[Kyle]

Huh. I checked. But my pants are clean. I'm stumped.

[Mike]

This is a type of logic puzzle I've only recently come to appreciate. It involves a hypothetical problem solver (the son) and then you have to use not only the information provided to the subject of the story, but then also use their assumed knowledge and process to figure out what they already know. For some reason, the first handful of these made sense to me once explained, but it took me a while to learn to solve them on my own. I just wasn't thinking the right way for them.

[Spoiler-ish hint follows:] So we assume the son knows his own age as well as his mother's. So when he looks at all the possible age combinations that multiply to 2450 and add to twice his own age, it still doesn't narrow it down for him. But somehow, knowing his mom's age combined with her cake comment helps him sort it out.

Here's another problem that I lump into this same category. It's not solved the same way, but it has that same idea of examining the subjects' viewpoints (and in this particular case, them examining each other's viewpoints) to know what they can learn and when. I've seen this video three different times, and I still don't know that I could solve it on my own.

[poorpete]

Are the three numbers....

50
49
1

assuming that they are part of Cory's family and thus the three people are Cory, the mom, and the son. The 2450 gets him some info, and the double his age gets him the three numbers. Then the final gets the information the Cory is younger than the mom.

Which makes the mom 50, if they are all members of the family though unsure given the last statement's use of "when." Then again, now that i think about it how can the 1 year old son talk so I'm probably wrong. Unless, Cory is 1 and the son is adopted, ¯\_(ツ)_/¯

[Mike]

I didn't realize that could be confusing. My apologies. I was trying to rephrase it from an older version involving a priest and cantor and three parishioners. There are five people in this puzzle: 3 members of "Cory's family", the mother, and the son. I will rephrase the original problem to clarify.

Using the given information, you can (will) deduce the ages of all five people, however, only the mother's age is required to answer the problem.

[poorpete]

I think I was searching for a trick that wasn't there

[Eliahad]

Okay, what's the answer?

[Tahlvin]

I know the answer, but I cheated a little because I didn't want to manually list out all the possible combinations of numbers. But I'll let Mike explain, since it's his question.

[Eliahad]

I mean, I think I figured out a lot of it, but I don't know if I'm right about some of my assumptions, and therefore don't trust my ability to write it down.

[Mike]

SOLUTION

First, you have to look at every combination of ages the three people could be. There are 20 total combinations, but some are ridiculous, because one person would have to be 250+ years old. (It helps here to break 2450 into it's prime factors of 2, 5, 5, 7, and 7 to figure out the possible combinations faster.)

Assume the son has done those calculations. He's looking for ones that sum up to twice his age. He says he still can't solve it, so that means there have to be two or more combinations that add up to twice his age.

So we look at the sums for all of them and look for different combos that add up to the same number. There are only two that fit the criteria: either Cory's family is 50, 7, and 7, or they are 49, 10, and 5. Both add up to 64, so we know the son is 32.

Then Mom says "when they reach my age," which implies she is older than they are, so she can't be 49 or under. But this statement also gives the son the answer, and if the mom was 51 or older, then he still wouldn't know. Therefore, she must be 50, meaning Cory's family is 49, 10, and 5.

[Eliahad]

Okay, I was on the right track, I got stuck on Cory's family being 35 and 35 and 2. Which doesn't help, because I didn't look to think of factors that created the same sum. Now I understand more.