Having failed to solve the problem mentioned in this post, the best I can do is throw up another puzzle (from the entertaining book "The Chicken from Minsk").

Two mathematicians, Igor and Pavel, met in the street. "How are you? How are your sons?" asks Igor. "You have three sons as I remember, don't you? But I have forgotten their ages". "Yes I do have three sons", replies Pavel. "The product of their ages is 36". Looking around and then pointing to a nearby house, Pavel says, "The sum of their ages is equal to the number of windows in the building over there." Igor thinks for a minute and then responds, "Listen, Pavel, I cannot find the ages of your sons." "Oh, I am very sorry," says Pavel; "I forgot to tell you that my eldest son has red hair." Now Igor is able to find the ages of the brothers. Can you?

## Thursday, 2 December 2010

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## 11 comments:

Neat. But, on second thoughts, strictly speaking this doesn't work: the story needs some tweaking. It also helps to know something about building regulations in Minsk :-)

I presume the "eldest son has red hair" thing is to inform us that there is only one occurence of the number of the greatest value... but i am no mathematician, so i'll leave it to others!

2,2,9

- Shanon

hmm and if 9, 2, 2 works, then so does 2, 3, 6 - so are there two possible correct answers? I guess (1, 1, 36) or (1, 2, 18) would probably be too many windows? I am thinking also that red hair in Minsk may be an oddity - perhaps that son is adopted?? We can rule out (1, 6, 6) I guess as TrueFiendish suggests.

Truefiendish is on the right track with his observation. Think about why extra information was required at each stage before an answer could be found.

The most likely scenario is that the eldest (red-haired) son is 6 years 11 months, the middle child has just turned six, and the baby is about eighteen months (6x6x1=36); that the building has 6+6+1=13 windows in the front and 25 in the sides and back; and the Matrons expect Igor and Pavel back at their respective institutions by six o'clock.

there are 13 windows.

Possible solutions were 6,6,1

or 9,2,2.

The eldest has red hair implies 9,2,2.

cheers Garrett.

Both Shannon and Garrett are correct although for an "A" full working needed to be shown.

This was a problem that was both about numbers and information. As Scott observed after the first piece of information there were many possible solutions. With the second piece of information (number of windows) the set of solutions was narrowed, but crucially for the problem not narrowed enough. This required a third piece of information (the eldest son) to provide a unique solution.

If you start with all the possible solutions you will find that they all have a unique sum of ages (ie 1,1,36 = 38, 1,2,18 = 21) with the exception of 2,2,9 and 1,6,6 which both equal 13. This is why Igor was not able to solve it after the second piece of information (eg if he could see 38 windows he would have known the answer straight away). Then, as Truefiendish pointed out, the third piece of information was required to distinguish between the two remaining answers.

(Note: While Ian Rout's answer may be clever, it completely kills the charm of the problem)

But Rout is right, as well as being clever, which is why I said that the story doesn't work.

First, there is a step before Shaun's in that we have to realise that the mathematicians (implausibly, if they are the pedants that they seem) are ignoring the elements of the childrens' ages that can't be expressed in integers. Whilst the ages could be 8, 4.5, and 1 (product 36)the number of windows could not be 13.5. (Although there is a thing called a half-window, when we count it it magically becomes one window, and not half a window.)

Second, the red hair is a red herring, as TrueFiendish said, to tell us that one of the children is the oldest. But that is quite compatible with Ian's solution. Even if the children were twins, one would have been born before the other, and it would be correct to call that child the eldest. Hence the new information doesn't allow Igor to distinguish between the two possible solutions.

The mistake in the story, a fairly elementary one, is to use a continuous variable (age) when it should have used a discrete variable - such as how many goldfish the children have, or how many chess tournaments they've won.

Hence I stick to my claim that it is neat, but needs rewriting.

For an "A" (and posterity) I will provide my working which is essentially as provided by previous posters:

The pertinent points of the problem can be restated as follows:

a. We wish to find a set of 3 integers with a product of 36,

b. knowledge of their sum is insufficient for a unique solution, and

c. knowledge that the set contains a unique largest element is sufficient for a unique solution.

From a. candidate solutions (with sums as shown) are as follows:

{36,1,1} sum = 38

{18,2,1} sum = 21

{12,3,1} sum = 16

{9,4,1} sum = 14

{9,2,2} sum = 13

{6,6,1} sum = 13

{6,3,2} sum = 11

{4,3,3} sum = 10

From b. we can remove possible solutions with a unique sum leaving:

{9,2,2},

{6,6,1}

From c. the unique solution is:

{9,2,2}

Shanon

...A blog mainly devoted to chess.

...yet with mathematical puzzle!

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