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Re: Table for FindInstance solutions
*To*: mathgroup at smc.vnet.net
*Subject*: [mg114594] Re: Table for FindInstance solutions
*From*: Daniel Lichtblau <danl at wolfram.com>
*Date*: Thu, 9 Dec 2010 06:02:53 -0500 (EST)
MH wrote:
> Hello. I'm trying to display in a table solutions to the equation 6S
> + 9N + 20T = D, where D is an integer allowed to go from 1 to 100. I
> can use FindInstance to find solutions with no problems. What'd I'd
> really like, though, is a table with four columns: one column each for
> D, S, N, and T. The following code gets me close to what I want, but
> every entry in the last three columns is the same. And, every entry
> gives the value for S, N, and T, instead of just the appropriate S or
> N or T value. For example, when D = 71, that row reads
>
> 71 {S -> 7, N -> 1, T -> 1} {S -> 7, N -> 1, T -> 1} {S -> 7, N ->
> 1, T -> 1}.
>
> I can interpret the correct result (7 sixes, 1 nine, and 1 twenty),
> but I'd like the column to read
>
> 71 7 1 1
>
> How can I change that? Incidentally, this problem is from a current
> issue of Delta Airlines' Sky Magazine. The problem asks about a baker
> who sells donuts in boxes of 6, 9, and 20. What's the largest number
> of donuts you CANNOT purchase? I have the solution; I'd just like it
> to look a little bit nicer. :-)
>
> Thanks!
>
> MH
>
> =====
> TableForm[
> Table[
> {D,
> FindInstance[
> 6 S + 9 N + 20 T == D && S >= 0 && N >= 0 && T >= 0, {S, N, T},
> Integers],
> FindInstance[
> 6 S + 9 N + 20 T == D && S >= 0 && N >= 0 && T >= 0, {S, N, T},
> Integers],
> FindInstance[
> 6 S + 9 N + 20 T == D && S >= 0 && N >= 0 && T >= 0, {S, N, T},
> Integers]
> },
> {D, 1, 20, 1}],
> TableHeadings -> {None, {"TOTAL", "BOXES OF 6", "BOXES OF 9",
> "BOXES OF 20"}},
> TableAlignments -> {Center},
> TableSpacing -> {1, 3}
> ]
> =====
That particular example is also know as the Chicken McNugget problem
(from the original sizes available for purchase of for McDonalds'
Chicken McNuggets). I guess the Delta Air publisher didn't want a
trademark infringement problem.
http://en.wikipedia.org/wiki/Coin_problem
Mathematica can compute this directly via FrobeniusNumber.
To answer your question, you could form the table as:
Table[Flatten[{d, ({s, n, t} /.
FindInstance[
6*s + 9 *n + 20*t == d && s >= 0 && n >= 0 && t >= 0, {s, n,
t}, Integers]) /. s | n | t -> ""}], {d, 1, 20, 1}]
Customary caveats:
(1) Generally a bad idea to start variable names with capitals.
(2) Always a bad idea if the variable is N, since that is a Mathematica
built-in function.
Daniel Lichtblau
Wolfram Research
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