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Today my girl-friend asked me to solve this problem. She knows that I am preparing for the GMAT and tries to help me by giving me hard logical problems to solve. This one I cracked in about 7-8 minutes. source: math competition problem for 5th grade kids. (well, it's debatable as for me).

Once two mathematicians (A and B) met each other. They had the following conversation: A:-Hi, how are you? B:-I am fine, thanks, I have two nice preschool sons. A:-Wou, great. How old they are? B:-Product of their ages equals to the number of birds sitting on this fence. A:-But this information is not sufficent. B:- ...and the smaller boy is similar to me. A:-Aha, I know their ages now. What are the boys' ages? Assume that numbers are integers.

I will post the answer later .

p.s. Please provide solution and not just a random guess.

Let: a- years of an older boy b- year of yanger boy Prescolar age is up to 6-7 years, well it even could be 5 years. Thus product of a*b could be any number from 1 (1*1) to 49 (7*7). N-number of birds on fence. Additional information as per DS question is: 1) smaller boy is similar to me 2) N does not equal to 0.

We know that a>b, and a*b=N. Considering that birds are on fence, thus neither a , nor b equals to 0. Put yourself in shoes of the "A" guy, who counted birds. He counted the number of birds, and than he compared that number to a product of any two numbers. In his mind, N is formed by some different pairs of figures: let say (a1;b1), (a2;b2), (a3;b3) -the combination of possible values is limited. (for example he counted 12 birds on fence, so possible sollutions are only two pairs 2;6, 3;4, and not 1;12. - but in this case there is no clue what are the boys' ages). After he understood that one of child is smaller, THIS FACT was sufficient for him to conclude about the ages. So, in his possible solutions (pairs of figures) were a pair of two equal numbers and a pair of non-equal numbers. Factor of two equal numbers is a perfect square. Perfects squares are 4,9,16,25,36,49 - up to 7.

Possible pairs are: 4- (2;2) and (4;1) 9- (3;3) and (9;1) 16-(4;4), (1;16), (2;8) From these pairs only first pair of figures has prescolar ages. So boys' ages are 4 and 1.

Re: Hard DS: Two russian mathematicians [#permalink]

Show Tags

29 Aug 2010, 04:47

ramgmat wrote:

Just giving it a shot! Dunno if it's right though....

Clue 1:- PRESCHOOL. So i assume they are below three years old and its been mentioned they are integers.

So the options are (1,2) (1,1) (2,2)

Clue 2: "The smaller boy is similar to me." This means one is older than the other

The answer could be only 1,2.

Good! I like you way of judgement, you are nearby.

But: 1.Why you assume that they are below 3 years old? what if one of them is 5,6 or even 7 years old? 2.In your options there are three different pairs (1,2) (1,1) (2,2), BUT product of this numbers gives us different number of birds, ie 2, 1 and 4 respectively. But number of birds is defined and it is N.

Re: Hard DS: Two russian mathematicians [#permalink]

Show Tags

30 Aug 2010, 21:05

Pkit wrote:

ramgmat wrote:

Just giving it a shot! Dunno if it's right though....

Clue 1:- PRESCHOOL. So i assume they are below three years old and its been mentioned they are integers.

So the options are (1,2) (1,1) (2,2)

Clue 2: "The smaller boy is similar to me." This means one is older than the other

The answer could be only 1,2.

Good! I like you way of judgement, you are nearby.

But: 1.Why you assume that they are below 3 years old? what if one of them is 5,6 or even 7 years old? 2.In your options there are three different pairs (1,2) (1,1) (2,2), BUT product of this numbers gives us different number of birds, ie 2, 1 and 4 respectively. But number of birds is defined and it is N.

thanks for kudos

Based on whatever information is posted till now, we can guess that age of the younger son is 1. Hence whatever is the age of the other son, their product is the same as the number of birds.

Hence if based on these two statements.

1) -- B:-Product of their ages equals to the number of birds sitting on this fence.

2) -- B:- ...and the smaller boy is similar to me.

Answer should be C.
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Re: Hard DS: Two russian mathematicians [#permalink]

Show Tags

06 Sep 2010, 12:08

Solution: Let: a- years of an older boy b- year of yanger boy Prescolar age is up to 6-7 years, well it even could be 5 years. Thus product of a*b could be any number from 1 (1*1) to 49 (7*7). N-number of birds on fence. Additional information as per DS question is: 1) smaller boy is similar to me 2) N does not equal to 0.

We know that a>b, and a*b=N. Considering that birds are on fence, thus neither a , nor b equals to 0. Put yourself in shoes of the "A" guy, who counted birds. He counted the number of birds, and than he compared that number to a product of any two numbers. In his mind, N is formed by some different pairs of figures: let say (a1;b1), (a2;b2), (a3;b3) -the combination of possible values is limited. (for example he counted 12 birds on fence, so possible sollutions are only two pairs 2;6, 3;4, and not 1;12. - but in this case there is no clue what are the boys' ages). After he understood that one of child is smaller, THIS FACT was sufficient for him to conclude about the ages. So, in his possible solutions (pairs of figures) were a pair of two equal numbers and a pair of non-equal numbers. Factor of two equal numbers is a perfect square. Perfects squares are 4,9,16,25,36,49 - up to 7.

Possible pairs are: 4- (2;2) and (4;1) 9- (3;3) and (9;1) 16-(4;4), (1;16), (2;8) From these pairs only first pair of figures has prescolar ages. So boys' ages are 4 and 1.[/spoiler]

Please explain....did not understand the colored part clearly..