2010/3/31 Kelly Jones said:
A classic math problem asks "using only four 4s and any
mathematical operation, come up with the number
56". "4!+4!+4+4" is one of probably several answers [1].
I want to write a ruby program that solves the
generalization: "given X number of the digit Y and a
set of permitted mathematical operations/functions, can
you come up with the number Z?"
As long as you do not limit the number of unary operations that
program is not guaranteed to terminate - especially if there is no
solution for given X, Y and Z. Am I missing something?
It should be interesting to see what solution strategy you are picking.
Thanks, and thanks to everyone else who replied.
You're right: I hadn't thought of this.
If I start with 4 and allow sqrt, I get sqrt(4),
sqrt(sqrt(4)), sqrt(sqrt(sqrt(4))) and so on. Of
course, these aren't integers after the first sqrt, but
they could theoretically combine with other 4
combinations later to form an integer.
I incorrectly thought one iteration of a unary operator would suffice.
I originally got interested in this problem because I
thought factorial was a cheat. Some of the solutions on
http://mathforum.org/ruth/four4s.puzzle.html use the
gamma function, integer 4th root, etc. Where do you
draw the line? If you allow constant functions (eg,
f(x) = 56), the solution is trivial.
My new goal is to solve the simpler problem, very
similar to
http://www.rubyquiz.com/quiz7.html (thanks,
Brian!).
Given X copies of the digit Y and the 5 mathematical
operators plus, minus, multiply, divide, and exponent,
along with concatenation and decimals (see below), can
you construct the number Z?
My approach for 5 copies of the digit 7 (example):
% With one 7, you have {7, 0.7} (the latter because we
allow decimals-- but not 0.07)
% With two 7s, we union two sets:
% {77, 7.7, .77} (from decimals and concatenation)
% Apply + - * / ^ to every ordered pair of elements
in the resultset for one 7 (including "pairs" like
{7,7}). Not showing the results, but you get the
idea.
% We then recurse. For n 7s, we union:
% {777...[n times], 777...[n-1 times].7, 777...[n-2 times].77, etc}
% Applying + - * / ^ to every ordered pair of the
resultset for n-1.
It might still be interesting to create a website that does this.
