       Re: Re:Playing with numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg50242] Re: [mg50223] Re:[mg50135] Playing with numbers
• From: DrBob <drbob at bigfoot.com>
• Date: Sun, 22 Aug 2004 00:19:53 -0400 (EDT)
• References: <200408210704.DAA24819@smc.vnet.net> <opsc3o91yziz9bcq@monster.cox-internet.com>
• Sender: owner-wri-mathgroup at wolfram.com

```And here's a solution that checks only 18 cases that, still, is no more difficult to analyze than before.

conditions = Simplify@{h - f == i, f*g == c, e == f*b, a + b == h, c - d == a}
{f + i == h, c == f g, e == b f, a + b == h, a + d == c}

(1) Because all nine symbols must be distinct, those involved in product or quotient equations -- b, c, e, f, and g -- cannot be 1. Nor can the sums c and h. Hence one of {a, d, i} must be 1 (three possibilities).

(2) Therefore, 2 <= g == c/f <=c/2 <= 4. The same argument applies to f and b, so {b,f,g} is a permutation of {2,3,4}.

If one of {a,d,i} is chosen to be 1 we can solve for the other five variables (other than that variable and b, f, g). This reduces the number of cases to

3*3!

18

chooseOne[x_] := {a, b, c, d, e, f, g, h, i} /. First@Solve[conditions, Complement[{a, b, c, d,
e, f, g, h, i}, {b, f, g, x}]] /. x -> 1
chooseOne /@ {a, d, i}

{{1,b,f g,-1+f g,b f,f,g,1+b,1+b-f},{-1+f g,b,f g,1,b f,f,g,-1+b+f g,-1+
b-f+f g},{1-b+f,b,f g,-1+b-f+f g,b f,f,g,1+f,1}}

solution[s_List, {bx_, fx_, gx_}] := Module[{sol = s /. Thread[{b, f, g} -> {bx, fx, gx}]},
If[Union@sol == Range@9, sol, {}]    ]
Flatten[Outer[solution, chooseOne /@ {a, d, i} //
Evaluate, Permutations@{2, 3, 4}, 1], 1] /. {} -> Sequence[]

{{5, 4, 6, 1, 8, 2, 3, 9, 7}}

Bobby

On Sat, 21 Aug 2004 16:14:15 -0500, DrBob <drbob at bigfoot.com> wrote:

> That's pretty cool! It looks at 45 cases, though; I have a solution that looks at 24, with a pre-coding analysis that's no more onerous (I think).
>
> conditions = {h - f == i, f*g == c, e == f*b, a + b == h, c - d == a};
>
> Note g == c/f <= c/2 <= 4. Also g >= 2 as otherwise c == f. The same argument applies to f and b. Hence {b,f,g} is a permutation of {2,3,4}.
>
> Given that, we try solving for the other six variables. But Solve can get only five (there are five equations).
>
> First[Solve[conditions, {a, c, d, e, i}]];
> simpleSolve = {a, b, c, d, e, f, g, h, i} /. %
> {-b + h, b, f*g, b + f*g - h, b*f, f, g, h, -f + h}
>
> Note that four of these depend on h. The sums h==f+i==a+b imply that h is strictly greater than a, b, i, and f. Since {b, f, g} is a permutation of {2,3,4}, at least one of a and i must be bigger than four. Hence h>=6.
>
> Therefore, the number of possible solutions reduces to:
>
> 3!4
>
> 24
>
> In particular, the possible values of {b, f, g, h} are:
>
> bfgh = Flatten[Outer[Flatten@{#1, #2} &, Permutations@{2, 3, 4}, Range[6, 9], 1], 1];
>
> Finally we check all 30 cases and list the solutions:
>
> check@{bx_, fx_, gx_, hx_} := Module[
>      {s = simpleSolve /. Thread[{b, f, g, h} -> {bx, fx, gx, hx}]},
>      If[Sort@s == Range@9, s, {}]]
> check /@ bfgh /. {} -> Sequence[]
>
> {{5, 4, 6, 1, 8, 2, 3, 9, 7}}
>
> That's a lot better than looking at 9!==362880 solutions!!
>
> Bobby
>
> On Sat, 21 Aug 2004 03:04:29 -0400 (EDT), Fred Simons <f.h.simons at tue.nl> wrote:
>
>> The question is to solve with Mathematica the set of equations
>> (1) H - F = I
>> (2) F x G = C
>> (3) E / B = F
>> (4) A + B = H
>> (5) C - D = A
>> where every variable from A to I stands for a number from 1 to 9 and each
>> number appears only once.
>>
>> Here is a solution where Mathematica does the computations that we otherwise
>> had to do by hand.
>>
>> Since the symbols represent different integers between 1 and 9, we conclude
>> from the equations i+f=h, a+b=h that h has to be at least 5. From the
>> equations c=f*g, e=f*b we conclude that f is between 2 and 4, that g is at
>> least 2, that e is at least 2, that c is at least 6 and that e is at least
>> 6.
>>
>> Given values for f, h, c and e, we can compute the other variables. The
>> result has to be a permutation of the numbers 1, ..., 9.
>>
>> In:= solution = {a, b, c, d, e, f, g, h, i} /.
>> Solve[{h - f == i, f*g == c, e/b == f,
>> a + b == h, c - d == a}, {a, b, d, g, i}][]
>>
>> Out=
>> {-((e - f*h)/f), e/f, c, c + e/f - h, e, f, c/f, h, -f + h}
>>
>> In:=
>> Timing[Do[If[Length[Union[solution]] == 9,
>> Print[solution]], {f, 2, 4},
>> {c, Ceiling[5/f]*f, 9, f}, {e, Ceiling[5/f]*f, 9, f}, {h, 5, 9}]]
>>
>>> From In:=
>> {5,4,6,1,8,2,3,9,7}
>>
>> Out=
>> {0. Second,Null}
>>
>> So indeed there is only one solution.
>>
>> Fred Simons
>> Eindhoven University of Technology
>>
>>
>>
>
>
>

--
DrBob at bigfoot.com
www.eclecticdreams.net

```

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