# "Child's Play" - A Logic / Math Puzzle

I thought that perhaps dauerbach and some of the other math/logic puzzle people might like this one:

Child’s Play

“Are those your children I hear playing in the yard?” the visitor asked his host.

“Actually, there are four families of children,” the host replied. “My family is the largest, my sister’s family is smaller than mine, my brother’s is smaller still, and my cousin’s family is the smallest. They’re playing Hide-and-Seek. They prefer baseball, but there aren’t enough children to field two full teams. Oddly enough, the product of the number of children in all four families is the same as my house number, which you saw when you came in.”

“I am something of a mathematician,” the visitor said. “Let me see if I can figure out how many children are in each family.” He puzzled over it for a few moments, and then said, “I need more information. Does your cousin’s family consist of a single child?”

His host answered his question, whereupon the visitor said: “Knowing your house number, and knowing the answer to my question, I can deduce exactly how many children are in each family.”

How many children are in each family?

Average time of solution:

Genius — 2 minutes
Bright — 20 minutes
Normal — Will give up after 20 minutes
Quarter-wit — Regrets ever coming in here, and will leave immediately

For those who don’t believe a solution is possible: Yes, there is a correct answer, and yes, you do have enough information to figure it out. There’s no trick; it merely requires a certain amount of logic and tenacity.

I emailed you my answer. I’d post but can’t find how to do a spoiler tag.

Argh! Never mind. I used faulty logic so I withdraw my answer. It fits the criteria but isn’t certain.

I corrected the flaw in my logic and emailed you my final answer. I believe I have the only combination that fits all the criteria.

I’m certainly no math whiz, but I do like puzzles.

I have the answer narrowed down to two possibilities. Is my reasoning correct? What am I figuring incorrectly that I can’t narrow it down to just one?

For those who still want to figure it out independently of my attempt, I will post my reasoning in a spoiler box:

[spoiler]OK… anyone who likes puzzles has seen other puzzles like the one they are trying to do, and this one is no exception. I’ve seen another puzzle where the product of something totaled an amount you are not told, but when the person in the story asked a follow-up question, was able to produce an answer, even though the reader was not told what that answer was. The trick was that the product could only be made up of a limited amount of factorials, and only one solution had (or didn’t have) a duplicate number in the factorials.

Applying the same logic here, we are told that the total number of children is less than two baseball teams; that is, the total number of children must be under 18: 17 or less. We are also told that each family has a different number of children, that is, no two have the same number. The smallest combinations of numbers that work would be 1, 2 3 and 4. That adds up to a total of 10. If the largest family was 6 children, followed by 5 then 4 then 3, that adds up to 18, which is too much. So the numbers must be less than that. The only numbers that are possible given these two limitations, then, are the following (with their products, which will be the next step):

6,5,4,2 = 240
6,5,4,1 = 120
6,5,3,2 = 180
6,5,3,1 = 90
6,5,2,1 = 60
6,4,3,2 = 144
6,4,3,1 = 72
6,4,2,1 = 48
6,3,2,1 = 36
5,4,3,2 = 120
5,4,3,1 = 60
5,4,2,1 = 40
5,3,2,1 = 30
4,3,2,1 = 24

Now, we are told that the product of the four numbers equal the house number (and we are not told what number that represents). But notice of all the products of the possible four numbers, only two sets have the same products, all others produce products that are different from all the others. If any of those were the correct numbers, the person in the story would know the answer right away (he has seen the house number, we are told). So it must be one of the two sets that produce duplicate products:

Both 6,5,4,1 and 5,4,3,2 have the same product of 120
Both 6,5,2,1 and 5,4,3,1 have the same product of 60.

Then we are told the person in the story had one last question: was the smallest family size 1 child? Although we are not told the answer, we know that provided the person in the story with enough information to deduce the correct answer. Of the two different possible sets, the second set both have 1 child, so the answer would not be helpful to narrowing down the two to one correct answer.

The other possible set is either 6,5,4,1 or 5,4,3,2. Knowing if the smallest family has one child would tell us which of the two is correct.

But as I don’t know the answer to that question, I cannot see how to narrow the two down to one myself.

Am I missing something simple? Am I completely off?[/spoiler]Padeye, on preview I see your posts. A spoiler box is easy, it’s just a tag before and after the spolier. the tag before is <spolier> (with the square brackets “[” instead of the “<”) and the end tag is </spoiler>

Well crap on a crutch. Logic was right but sloppy arithmetic. Here’s my final, final answer with my logic. Coincidentally it was the first answer I emailed you.

2,3, 4 and 5 This presumes the mathemetician did not count the total number of childer. If he had he would have already known there were too few for a baseball game. it is one of three combinations that have a product of 120, the street number, and the only one with two children in the smallest family. The other two factor sets are 1, 4, 5 and 6 and 1, 3, 5, and 8.

Interesting. But not hard at all.

The visitor knows the house number. There are only 38 sets of four positive unique (each family size is different) integers that satisfy the conditions of having a sum less than 18 (not enough to field two baseball teams). Therefore, the visitor should have known immediately what the answer was… unless the house number happened to be one for which multiple sets gave the same product. Funny how that happens in puzzles like this!

In any event, 48 is the product of two sets, 60 is the product of three sets, 72, 80, 84, 90, and 96 are all the product of two sets, and 120 is the product of three sets. The house number must be one of those, or else the visitor would have already given his answer.

This shows why he asks the question about the cousin’s family having a single child. If the house number were 60, for example, this information would be of no use: all three sets that produce 60 have the cousin’s family with a single child.

However, if the house number were 120, the question would help: only one of those sets has a cousin family of size 2.

Therefore, the house number is 120, and the cousin has two children, the brother three, the sister four, and the host five. That makes their sum fourteen, too small to field two baseball teams.

• Rick

I must have misspelled something the first time I tried to preview a spoiler tag but have my answer and logic and I’m satisfied with it.

Questions for Bricker:

How did you know 48, 60, 72. 80, 84, 90, 96 and 120 are:

1. products of multiple sets
2. the only numbers which are products of multiple sets for which the factors sum to less than 18?

How did you know there are 38 unique sets?

I had to do all that manually before solving it. I ended up programming it and looking for the numbers that way. I think I missed a mathemetica theorem which would have helped tremendously.

Thanks

Yeah, I’d like to know, also. Let’s see…

[spoiler]cmosdes, I don’t know if Bricker had a mathematical way of doing it, but I counted them out. However, I only counted 37 unique sets, and only 2 products were 60 (not 3). Obviously, if I’m missing one, it’s product is 60.

Other than that, I got the same solution through roughly the same method. By the way, I just stuck the numbers in a spreadsheet, sorted by sum, and removed all of those whose sum was 18 or higher.
[/spoiler]
Thanks for the puzzle, Dr. Cobweb.

Oops. I found the one that was missing, there were 38.

This puzzle assumes the kids are not ardent supporters of the DH rule!

I’d like to have a go, but I have no idea how many players are in a baseball team. Searching on Google just brings up a bunch of sites that assume you at least have a basic idea of the game.

Could someone enlighten me?

Thanks

9

MonkeyMensch is correct; puggyfish, a standard baseball team fields 9 players, as El Smasho has indicated.

Padeye and Bricker seem to have worked this type of puzzle before; they both have the logic down. (And thanks to all for using <spoiler> tags to keep things interesting for others.) For those who are interested but stuck and impatient, here’s a detailed description of the approach:

[spoiler]First, note that there must be less than 18 children. Otherwise, they would be able to field two full baseball teams. This limits the possibilities. Also, note that the number of children in each family increases by at least one for each family, going up from the cousin’s, to the brother’s, to the sister’s and finally to the host’s. These are the first significant facts.

Now, note that the cousin’s family (the smallest) cannot have more than 2 children. If it did, the smallest possible numbers would be 3, 4, 5, 6. But that adds up to 18, which we know is too many children. So the only possibilities for the cousin’s family are 1 and 2.

(Mathematical purists will consider the possibility of 0 children in the cousin’s family. This can be eliminated because it would mean the host’s house number would have to be 0 also, and there is no way the visitor could resolve the problem in that case, since all products would yield the same result. Yet he did resolve it. Therefore, 0 children in the cousin’s family is ruled out.)

It is now possible (if a bit tedious) to list all the possibilities for the numbers, from 1, 2, 3, 4 to 2, 4, 5, 6. See the table below for a complete list. Note that we divided the list into two columns, starting with 1 or 2 (representing the cousin’s family).

Now, consider the visitor’s statement and question: “I need more information. Does your cousin’s family consist of a single child?” The fact that he had to ask this question is the key to the puzzle, and it tells us two important things:

1 - The house number occurs more than once in our list of solutions. Had the solution been unique (i.e., if there were only one set of four numbers whose product was the house number), he would never have had to ask this question.

2 - The house number must occur in both columns of our table. If it didn’t, the visitor would not have needed to ask this particular question, because the answer wouldn’t tell him anything new. The only reason for him to ask this question is to determine which of the two columns the correct answer is in.

We now look back to our table of possibilities. There is only one product (house number) that occurs in both columns: 120. It is the product of three combinations of numbers:

1 x 3 x 5 x 8 = 120
1 x 4 x 5 x 6 = 120
2 x 3 x 4 x 5 = 120

## And now we have the solution. If the host had answered “Yes” to the visitors question “Does your cousin’s family consist of a single child?”, the visitor still wouldn’t know the answer, because it could be either 1, 3, 5, 8 or 1, 4, 5, 6. Yet the visitor did know the answer. So the host must have responded “No”, and the only solution left is: Cousin’s family: 2 Brother’s family: 3 Sister’s family: 4 Host’s family: 5

TABLE OF POSSIBLE SOLUTIONS:
1 x 2 x 3 x 4 = 24
1 x 2 x 3 x 5 = 30
1 x 2 x 3 x 6 = 36
1 x 2 x 3 x 7 = 42
1 x 2 x 3 x 8 = 48
1 x 2 x 3 x 9 = 54
1 x 2 x 3 x 10 = 60
1 x 2 x 3 x 11 = 66
1 x 2 x 4 x 5 = 40
1 x 2 x 4 x 6 = 48
1 x 2 x 4 x 7 = 56
1 x 2 x 4 x 8 = 64
1 x 2 x 4 x 9 = 72
1 x 2 x 4 x 10 = 80
1 x 2 x 5 x 6 = 60
1 x 2 x 5 x 7 = 70
1 x 2 x 5 x 8 = 80
1 x 2 x 5 x 9 = 90
1 x 2 x 6 x 7 = 84
1 x 2 x 6 x 8 = 96
1 x 3 x 4 x 5 = 60
1 x 3 x 4 x 6 = 72
1 x 3 x 4 x 7 = 84
1 x 3 x 4 x 8 = 96
1 x 3 x 4 x 9 = 108
1 x 3 x 5 x 6 = 90
1 x 3 x 5 x 7 = 105
1 x 3 x 5 x 8 = 120
1 x 3 x 6 x 7 = 126
1 x 4 x 5 x 6 = 120
1 x 4 x 5 x 7 = 140
2 x 3 x 4 x 5 = 120
2 x 3 x 4 x 6 = 144
2 x 3 x 4 x 7 = 168
2 x 3 x 4 x 8 = 192
2 x 3 x 5 x 6 = 180
2 x 3 x 5 x 7 = 210
2 x 4 x 5 x 6 = 240
[/spoiler]

I think I fit into the fifth category.

Riddles you already know the answers to go in MPSIMS. I’ll move this over there.

bibliophage
moderator GQ

richardb, here’s what you missed out… I’ve added information to complete your logic and determine the answer:

You have ignored combinations with the largest family having more than 6 children, such as:
10, 3, 2, 1 which also gives 60.

And for your other question of eliminating using the information from the last question, you came to:

Both 6,5,4,1 and 5,4,3,2 have the same product of 120
Both 6,5,2,1 and 5,4,3,1 have the same product of 60.
However, you had discounted the possibility of 8, 5, 3, 1 = 120.

This would make the answer relevant and decisive only if the answer received for the last question was 2.

Therefore, using your logic, plus this information, the answer would be 120 children, with families of 5,4,3,2 children.

small typo:

… the answer would be house number 120, with families of 5,4,3,2 children.

Thanks, xash, but I figured that out already.

The first post after mine, by Padeye, pointed out the possibility of having more than 6 kids in a family and still totalling less than 18. It seems obvious now, but this is not the first time (nor probably the last) that I’ve made a stupid mistake.

If that had been the only post I read, then perhaps I would have tried again, but Bricker’s post immediately after also provided the solution.

At least I can comfort myself with my belief that I had the logic figured out correctly, and if not for that stupid mistake I would have been the first to post the correct answer!