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Thu 30 Nov, 2006 11:20 am

Can anyone help me solve this question??

You are confronted with the following formula:

A * [(B + C)(D - E) - F(G*H) ] / J = 10

Knowing that each variable is a unique, single-digit, nonzero number, and that C - B = 1, and H - G = 3, what is the number ABCDEFGHJ, where each letter is a digit? For example, if A = 4, B = 2, and C = 7, ABC would equal 427.

A * [(B + C)(D - E) - F(G*H) ] / J = 10

interesting.

this is tough

wait. each letter is a variable to a SINGLE-DIGIT, NON-zero number.. there are 10 letters. Say what?

A=

B=

C=

D=

E=

F=

G=

H=

I=

J=

Brute Force Solution (non-unique results)

Actually there are only 9 letters. For some reason the formula skips I and goes to J.

In any case, you can build a matrix of possibilities where every letter could be one of nine possible digits. Then just start eliminating possibilities using principles like:

H = G + 3. so H cannot be 1,2, or 3 (because that would make G a zero or negative number) and likewise G cannot be 7,8 or 9

One significant hint is that J is the only divisor. Since the final result (10) is a whole number, J must be evenly divisible into the rest of the formula. If you break it down, this also means J must be evenly divisible into the product of A*F*G*H. Thus, J cannot be 5 or 7 since no product of any other numbers could yield a multiple of 5 or 7 without using 5 or 7 themselves.

You can do other tricks like ensuring that (B+C)(D-E) > FGH to ensure a positive result. Honestly though, at this point in the problem I was getting restless so I decided to be cheap and just whip up a Python script to try all 7! possibilities by brute force (only 7! and not 9! since the C-B=1 and G-H=3 offer a direct dependent relation between 2 variables). This is only about 5,000 possibilties so it actually ran within a few seconds and it turns out there are 3 non-unique solutions:

[A, B, C, D, E, F, G, H, J]

[2, 6, 7, 8, 3, 5, 1, 4, 9]

[2, 6, 7, 9, 5, 3, 1, 4, 8]

[6, 7, 8, 4, 1, 3, 2, 5, 9]

lillypopins wrote:You are confronted with the following formula:

A * [(B + C)(D - E) - F(G*H) ] / J = 10

Is there any particular reason you didn't post this to

**the Neopets "Lenny" topic**? Did you have some objection to

**the solution** posted there...?

Thank you.

Eliz.