# Ruby solve classic math problem: make 56 from four 4s

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?”

Has anyone already done this?

On Thu, Apr 1, 2010 at 6:11 AM, Kelly J.
[email protected] wrote:

The first thing that came to mind was
http://www.rubyquiz.com/quiz60.html
Not exactly what you want, but still good reading

2010/3/31 Kelly J. [email protected]:

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.

Kind regards

robert

Michael F. wrote:

The first thing that came to mind was
Ruby Quiz - Numeric Maze (#60)
Not exactly what you want, but still good reading

Even closer is this one:
http://www.rubyquiz.com/quiz7.html

At 2010-03-31 05:11PM, “Kelly J.” wrote:

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?”

Has anyone already done this?

See http://rosettacode.org/wiki/24_game_Player
This uses 24 instead of 56, and the operations are limited to ±*/
but the idea is there.

On 4/1/10, Robert K. [email protected] wrote:

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
Classroom Resources - National Council of Teachers of Mathematics 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 Ruby Quiz - Countdown (#7) (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.