January 24th, 2014 at 6:45:03 PM
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Quote:AxiomOfChoiceAren't there infinitely many solutions to that?

Yes - in fact, I'm pretty sure that there is a solution P(x) = ax

^{3}+ bx

^{2}+ cx + d for every integer a.

I used a differences method on P(29) - P(28) and P(28) - P(27) to get:

216 x

^{2}- 12013 x + 167020
January 25th, 2014 at 6:42:00 AM
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Good answer, Don.

The beauty of the problem, I think, is in the answer NOT requiring to know the P(x).

Bravo, Don!

Indeed, one of many possibilities. Anyway, each P(x) is valid up to a multiplicative constant.Quote:ThatDonGuyYes - in fact, I'm pretty sure that there is a solution P(x) = ax

^{3}+ bx^{2}+ cx + d for every integer a.

I used a differences method on P(29) - P(28) and P(28) - P(27) to get:216 x^{2}- 12013 x + 167020

The beauty of the problem, I think, is in the answer NOT requiring to know the P(x).

Bravo, Don!

Reperiet qui quaesiverit

January 25th, 2014 at 7:02:57 AM
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A general solution

For every polynomial P(x) with integer coefficients, it is true that

P(a) - P(b) = (a-b) Q(a,b) where Q(x,y) is a polynomial with integer coefficients.

Knowing the value of P(a) - P(b), you can infer that (a-b) is a factor of it.

Using Don's notation (X=her age, M=Magician's guess, Y=Your guess)

P(M) - P(X) = 299 = 13 x 23 of which (M-X) is a factor (1, 13, 23, or 299)

P(X)- P(27) = 133 = 7 x 19 of which (X - 27) is a factor (1, 7, 19, or 133)

P(M) - P(27) = 166 = 2 * 83 of which (27 - M) is a factor (1, 2, 83, or 166)

This all put together has only two solutions, of which one respects the information that M was greater than X.

P(a) - P(b) = (a-b) Q(a,b) where Q(x,y) is a polynomial with integer coefficients.

Knowing the value of P(a) - P(b), you can infer that (a-b) is a factor of it.

Using Don's notation (X=her age, M=Magician's guess, Y=Your guess)

P(M) - P(X) = 299 = 13 x 23 of which (M-X) is a factor (1, 13, 23, or 299)

P(X)- P(27) = 133 = 7 x 19 of which (X - 27) is a factor (1, 7, 19, or 133)

P(M) - P(27) = 166 = 2 * 83 of which (27 - M) is a factor (1, 2, 83, or 166)

This all put together has only two solutions, of which one respects the information that M was greater than X.

Reperiet qui quaesiverit

January 25th, 2014 at 12:07:24 PM
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And here's a generic solution for P(x) as a cubic:

Let P(x) = ax

P(29) = 24389a + 841b + 29c + d = 299

P(28) = 21952a + 784b + 28c + d = 0

P(27) = 19683a + 729b + 27c + d = 133

P(29) - P(28) = 2437a + 57b + c = 299

P(28) - P(27) = 2269a + 55b + c = -133

(P(29) - P(28)) - (P(28) - P(27)) = 168a + 2b = 432

For any value a:

b = 216 - 84a

c = 299 - 2437a - 57 * (216 - 84a)

= 299 - 2437a - 12312 + 4788a

= 2351a - 12013

d = -21952a - 784b - 28c

= -21952a - 784 * (216 - 84a) - 28 * (2351a - 12013)

= a * (-21952 + 784 * 84 - 28 * 2351) + (-784 * 216 + 28 * 12013)

= -21924 a + 167020

P(x) = ax

If a is an integer, then so are the other three coefficients

Let P(x) = ax

^{3}+ bx

^{2}+ cx + d

P(29) = 24389a + 841b + 29c + d = 299

P(28) = 21952a + 784b + 28c + d = 0

P(27) = 19683a + 729b + 27c + d = 133

P(29) - P(28) = 2437a + 57b + c = 299

P(28) - P(27) = 2269a + 55b + c = -133

(P(29) - P(28)) - (P(28) - P(27)) = 168a + 2b = 432

For any value a:

b = 216 - 84a

c = 299 - 2437a - 57 * (216 - 84a)

= 299 - 2437a - 12312 + 4788a

= 2351a - 12013

d = -21952a - 784b - 28c

= -21952a - 784 * (216 - 84a) - 28 * (2351a - 12013)

= a * (-21952 + 784 * 84 - 28 * 2351) + (-784 * 216 + 28 * 12013)

= -21924 a + 167020

P(x) = ax

^{3}+ (216 - 84a) x

^{2}+ (2351a - 12013) x + (167020 - 21924a)

If a is an integer, then so are the other three coefficients

January 25th, 2014 at 12:27:35 PM
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Did we ever find out old she was?

January 25th, 2014 at 12:35:23 PM
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So you're saying that she's 28 and the Magician's guess was 29. That seems to be the only answer that satisfies all the criteria and your expanded formulae. Phooey, kind of; 1 did not seem valid to me in trying to suss it out. Oh, well.

If the House lost every hand, they wouldn't deal the game.

January 25th, 2014 at 1:30:56 PM
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Babs is at that awkward age. Old enough to know better, too young to resist the temptation.

Shed not for her
the bitter tear
Nor give the heart
to vain regret
Tis but the casket
that lies here,
The gem that filled it
Sparkles yet