The answer to the Cihan Altay puzzle is {A,B,C}={27, 8, 4}
Dr. Jaime Rangel-Mondragón
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I found a solution: A=27, B=8, C=4!
The proof:
1. A: "Is your number a perfect square?"
B: "No"
C: "Yes"
2. B: "Is your number greater than 50?"
A: "No"
C: "No"
3. C: "Is your number a perfect cube?"
A: "Yes"
B: "Yes"
possible perfect squares: 4,9,16,25,36,49,64,81,100
possible perfect cubes: 8,27,64
It has to be A>B>C
A knows that B has got number 8 because it is the only perfect cube that is
smaller than his own number (27).
A knows that C has got number 4 because it is the only perfect square smaller
than B's number (8).
B knows that A has got number 27 because it is the only perfect cube smaller
than 51 and greater than his own number (8).
B knows that C has got number 4 because it is the only perfect square smaller
than his own number (8).
C knows that A has got number 27 and that B has got number 8 because 27 and 8
are the only perfect cubes smaller than 51.
I did a computer search to proof that this is the only solution.
Jan Zernisch
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The numbers are 27, 8, and 4. This is why:
For this, we will call A's number a, B's number b, and C's number c.
1) It should be unnecessary to point this out, but for the sake of
being thorough let it be known that A knows a=27, B knows b=8, and C
knows c=4.
2) Remember that the set of possible numbers ranges from 2 to 100.
The number 1 was excluded from the set, and is therefore ignored in all
the statements below.
3) Remember that the mathematicians ask questions in an order based
on their numbers. The person with the highest number, A, went first.
The person with the lowest number, C, went last. Therefore a>b>c.
4) A goes first and asks "Is your number a perfect square?" B says
"no." C says "yes."
5) B goes next and asks "Is your number greater than 50?" A says
"no." C says "no."
6) C goes last and asks "Is your number a perfect cube?" A says
"yes." B says "yes."
7) A knows that b is a perfect cube less than a. Since the only
perfect cube less than 27 is 8, A knows b=8.
8) A knows that c is a perfect square less than b. Since the only
perfect square less than 8 is 4, A knows c=4.
9) B knows that a is a perfect cube less than or equal to 50. The
only two perfect cubes less than or equal to 50 are 27 and 8. Since B
knows b=8, B knows a=27.
10) B knows that c is a perfect square that is less than b. Since
the only perfect square less than 8 is 4, B knows c=4.
11) C knows that a and b are perfect cubes less than or equal to 50.
The only two perfect cubes less than or equal to 50 are 27 and 8. Since
a>b, C knows a=27 and b=8.
Clinton Weaver
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Each mathematician has answered only two questions about his number, one
of them about if it is either a square or a cube. For each mathematician
to know the others' numbers, they must all be squares or cubes. Since A
didn't reveal if his number was a square, and C didn't say if his was a
cube, A's number is a cube, C's is a square, and B's is either a square
or a cube (possibly both).
The only possible cubes for A's number to be are 64, 27, and 8. If A's
number is 64, then B's (smaller) number must be no more than 50, since
there are no squares or cubes between 51 and 63. Therefore C's (even
smaller) number would also have to be no more than 50, so the only other
info on C's number would be that it is a square. But the possibilities
for C's number are 4, 9, 16, 25, 36, and 49, and there's no way A could
know which it is; if B's number is a cube it could be either 8 or 27,
with C's number 4, 9, 16, or 25, and if B's number were a square there
would be even more possibilities. So A could not know B's and C's numbers
with certainty if his number was 64.
If A's number is 8, C's (smaller) square number would have to be 4, and
B's (between them) couldn't be either a square or a cube. Thus A's number
is 27.
B's number cannot be both a square and a cube, for it is smaller than A's
and 64 is the only number between 2 and 100 which is both. So it is
either only a square or only a cube. If it's only a square, it would have
to be 9, 16, or 25, with C's (smaller) number 4, 9, or 16, and A has no
way of knowing which it is.
Thus B's number is only a cube, and must be 8. C's (smaller) square
number is 4.
To prove that 27, 8, and 4 works, A knows that B's number is a cube and
under 27, so must be 8 (the only possibility), and C's must be 4 (the
only square under 4). From B's perspective, A's number must be a cube and
no greater than 50, so would have to be 27 (8 is his number). C's is a
square and smaller than 8, so must be 4. From C's perspective, A's number
is a cube and no greater than 50, so must be 27 or 8. B's number is a
cube and no more than A's, so can only be 8, meaning A's is 27.
Darrel C Jones
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> [...] the questions in order are:
> A: Is your number a perfect square?
> B: Is your number greater than 50?
> C: Is your number a perfect cube?
And the answers are:
(A's question) B: no, C: yes
(B's question) A: no, C: no
(C's question) A: yes, B: yes
The numbers themselves are A: 27, B: 8, C: 4. A knows that B's number is
a perfect cube smaller than his own, and that C's number is a perfect square
smaller than B's; the only possible choices are thus 8 and 4. B knows that
A's number is a perfect cube, less than 50 but greater than his own number;
the only one that fits is 27. Likewise, C's number is a square smaller than
8, which must be 4. And finally, C knows that Both A and B are perfect cubes
and that A's number is smaller than 50, which forces them to be 27 and 8.
Nifty puzzle!
Steven Stadnicki
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Hi Ed,
The answer to Cihan Altay's puzzle is 27, 8, 4.
Daniel Scher
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A = 27
B = 8
C = 4
A knows that B is a cube and is less than 27, so B=8. He also knows
that C is a square less than 8, so C=4.
B knows that A is a cube less than 50 and greater than 8, so A=27. He
also knows that C is a square less than 8, so C=4.
C knows that A and B are both cubes and A is less than 50, so A=27 and
B=8.
Nathan Stohler
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A: 27
B: 8
C: 4
After the first question, B knows C's number is 4.
After the second question, everyone knows A's number is less than 50.
After the third question. B knows A has 27. A knows B has 8. C knows both
of these. A, knowing B has 8, determines C has 4.
Ken Duisenberg
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Anupam 27,8,4