[superprismatic]

*** Inspired by CrayolaSunset's nice puzzles ***

For my 1,000^th post, here's one that I think is really hard because I think it

requires several insights which don't seem to pop to mind (at least not mine):

The letters of the alphabet used in this puzzle were each assigned a unique integer in the

range 2 to 26 such that no letter is assigned a value of a prime larger than 13.

Words are given values that are the product of the values assigned to the letters.

For example, if P=2, A=3, and T=5, then PAT = 30.

If FINK - (AQUEOUS)^(1/4) - (PILLS)^(1/3) - (BUTTER)^(1/2) = 154

(roots are all positive integers), then what are the values of FINK, AQUEOUS, PILLS,

and BUTTER?

Edited September 17, 2011 by superprismatic
"unique integer" replaces "number" in the first paragraph of the problem statement

[Guest]

We just have to notice that 154 = 250 - 16 - 16 - 64

So one solution is

F = 5, I = 2, N = 5, K = 5, FINK = 250

A = 16, Q = 4, U = 4, E = 4, O = 4, S = 4, AQUEOUS = 65536, (AQUEOUS)^(1/4) = 16

P = 8, L = 8, PILLS = 4096, (PILLS)^(1/3) = 16

B = 4, T = 4, R = 4, BUTTER = 4096, (BUTTER)^(1/2) = 64

[superprismatic]

We just have to notice that 154 = 250 - 16 - 16 - 64

So one solution is

F = 5, I = 2, N = 5, K = 5, FINK = 250

A = 16, Q = 4, U = 4, E = 4, O = 4, S = 4, AQUEOUS = 65536, (AQUEOUS)^(1/4) = 16

P = 8, L = 8, PILLS = 4096, (PILLS)^(1/3) = 16

B = 4, T = 4, R = 4, BUTTER = 4096, (BUTTER)^(1/2) = 64

Indeed, you are correct. I had erred in my problem statement. I meant to have the

puzzle have the feel of CrayolaSunset's problems in that different letters are assigned

different number values. This is what makes the problem difficult. Sorry. I will amend

the problem statement.

[bushindo]

Thanks for the problem. By the way, congrats on the 1000th post.

There are lots of solutions. Here are 3 of them

	 p  i  l  s  b  u  t  e  r  a  q  u  e  o  s  f  n  k

	 21  2 14  9 10  5 20  6 12 25 24  5 16  6  9  4  7 26

	 7 16 14  8  3 15 11 20 25 18  5 15  4 20  8  2  6 10

	 6 20  3 25 13  8 21  9 26 18 10  8  5  9 25  2  4 22

[Guest]

We just have to notice that 154 = 250 - 16 - 16 - 64

So one solution is

F = 5, I = 2, N = 5, K = 5, FINK = 250

A = 16, Q = 4, U = 4, E = 4, O = 4, S = 4, AQUEOUS = 65536, (AQUEOUS)^(1/4) = 16

P = 8, L = 8, PILLS = 4096, (PILLS)^(1/3) = 16

B = 4, T = 4, R = 4, BUTTER = 4096, (BUTTER)^(1/2) = 64

Could you explain how did you get the answer.

I have no idea to the puzzle at all.

Thanks a lot

[superprismatic]

Thanks for the problem. By the way, congrats on the 1000th post.

There are lots of solutions. Here are 3 of them

p i l s b u t e r a q u e o s f n k
21 2 14 9 10 5 20 6 12 25 24 5 16 6 9 4 7 26
7 16 14 8 3 15 11 20 25 18 5 15 4 20 8 2 6 10
6 20 3 25 13 8 21 9 26 18 10 8 5 9 25 2 4 22

Nice! I wasn't sure how many solutions there were. I was hoping that it was unique because I thought there

were enough constraints to do that. After I made up the problem, I solved it by hand. I never wrote a program

to look for answers.