Board 8 > Math Riddle

(+) Yeah! (+)

How many children?

Math Professor Smith explained to one of his graduate students (Jones) that one day his children were playing in Smith's backyard with the children from 3 other families.

Smith's family has the most children; the Browns have a smaller number of children; the Greene's have still a smaller number, and finally the White's have the smallest of all.

"How many children are there altogether," asked Jones.

"There are fewer than 18, and the product of the numbers in the four families happens to be my house number (which Jones already knew), said Smith.

Jones began to scribble some numbers. A moment later he said he needed more info: "is there more than one child in the White family?"

As soon as Smith replied, Jones knew the correct total of each family.

The number of children in each family can now be solved however based solely on the info provided!

Matt Riddle
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Matt Ridley

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I must be missing something, but I don't think that's solvable with just that info? Not without knowing the answer to that question.

The question being asked provides the info that there exists multiple combinations that give the same product (the unknown to us house number). Asking if the smallest family is greater than 1 tells you that you can reach it with the smallest family at 1 or 2.

The only overlap of products I can think of offhand that fit this are 2,3,4,5 and 1,4,5,6 which both yield a product of 120. Certainly hearing the answer to the presented question would eliminate one or the other but without hearing the answer ourselves... I don't think we can determine it?

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I was going to give up but then I remembered that one riddle from the "elementary school test in Japan" or whatever it was, and revisited it. Plus I'm a sucker for these things.

If you draw out all the combinations where: a + b + c + d < 18, and a < b < c < d

The fact he has to ask whether the Whites have 1 or 2 children, while already knowing the house number, means the "abcd" product has to occur in both a White = 1 and White = 2 situation. The only product that occurs with, is 120:

1, 3, 5, 8 or 1, 4, 5, 6 or 2, 3, 4, 5

And the only way he could know for sure after being told by the prof is if White = 2 so it was 2, 3, 4, 5

Ridley Scott

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oh duh i missed a combo. the above is right

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Thats what I got!

The part about the product was a semi-red herring. Its not so much needing to know what the product is, its more so knowing that Jones was aware of an isolated number (that happened to be the product).

Yeah I was two seconds away from asking if we were supposed to know how many children Smith has or what his house number was.

And... I literally jotted all the possible combinations in Excel so I mean.... more credit to anyone who gets there without going through all that.

I figured it out, got the same answer as Martin. The trick to the puzzle for me: the fact that we had enough info after he asked about the Whites. The whites must have 1 or 2, because if they had 3 there would be a minimum of 3+4+5+6 children, or 18. If the product was reachable in two ways, one with 1 and one with 2, we would gain no info from the last question - we would have no way of knowing what the answer was. But if there were three ways, two with 1 and one with 2, then we know by the fact that there were no further questions that he must have said "yes" - if he said no he wouldn't have had enough information. That's the only way for us to know the answer to that question.

Paratroopa1 posted...

I figured it out, got the same answer as Martin. The trick to the puzzle for me: the fact that we had enough info after he asked about the Whites. The whites must have 1 or 2, because if they had 3 there would be a minimum of 3+4+5+6 children, or 18. If the product was reachable in two ways, one with 1 and one with 2, we would gain no info from the last question - we would have no way of knowing what the answer was. But if there were three ways, two with 1 and one with 2, then we know by the fact that there were no further questions that he must have said "yes" - if he said no he wouldn't have had enough information. That's the only way for us to know the answer to that question.

And then from there i just tried a bunch of stuff until I got an answer that fit.

Jakyl25 posted...

Matt Riddle

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Paratroopa1 posted...

I figured it out, got the same answer as Martin. The trick to the puzzle for me: the fact that we had enough info after he asked about the Whites. The whites must have 1 or 2, because if they had 3 there would be a minimum of 3+4+5+6 children, or 18. If the product was reachable in two ways, one with 1 and one with 2, we would gain no info from the last question - we would have no way of knowing what the answer was. But if there were three ways, two with 1 and one with 2, then we know by the fact that there were no further questions that he must have said "yes" - if he said no he wouldn't have had enough information. That's the only way for us to know the answer to that question.

Yeah, this would be the simplest deduction from my incomplete post and in fact could be logically derived from it with the fact that 2,3,4,5 is the smallest possible combination that fits the criteria if White = 2. So necessarily any other White = 2 combination would yield a higher (and more relevantly to the question, different) product. Therefore, even if I missed a combination it would have to be in the White = 1 column upon which the rest of the answer falls into place that it must be the White = 2 scenario regardless of the fact that I couldn't provide the third valid combination. Just... I couldn't be sure that was what I was missing and that I wasn't overlooking some other information in the overall scenario.

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Its very fascinating how some riddles seem so obviously unsolvable until you delve into them. For me, this was one of them.

i have 6, 5, 4, 1 or 5, 4, 3, 2? but are there alternative solutions?
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SeabassDebeste posted...

i have 6, 5, 4, 1 or 5, 4, 3, 2? but are there alternative solutions?

Yeah, you missed the one I did 1, 3, 5, 8. So with 2 answers at W=1 and 1 at W=2, the fact that he immediately knew the solution upon hearing an answer to the "is there more than 1?" means it had to be the W=2 solution otherwise it would still be uncertain.

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Jakyl25 posted...

Matt Riddle

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I wrote up my summary before reading spoilers. Nothing new here but if you feel like seeing how my brain puzzled it out:

I'll refer to sets of 4 numbers meeting the stated criteria as "possible solutions".

The big clue is that knowing the answer of whether or not W = 1 will lead us to the solution. Therefore, one of these statements is true:

A) There is exactly one possible solution containing a 1, and there are multiple solutions which do not contain a 1.

or

B) There is exactly one possible solution which does not contain a 1, and there are multiple solutions which contain a 1.

In either case, knowing the answer to "does W = 1" will identify the answer.

Note that the product cannot have exactly two possible solutions, (one containing a 1 and one without a 1) because then while Jones could solve the riddle, we could not).

Since either A or B must be true, that means that at least one possible solution does not contain a 1. Therefore, the minimum possible value for the product would be 2 * 3 * 4 * 5 = 120. What possible solutions exist which contain a 1 and whose product is 120?

A little guess and check leads me to 1 * 3 * 5 * 8 and 1 * 4 * 5 * 6.

Thus, since I know there cannot possibly be a different solution not containing a 1 whose product is 120, and there are multiple possible solutions containing a 1 whose product is 120, I conclude that 2, 3, 4, 5 is the solution and thus the answer is 14.