Le Monde puzzle [#842]

An easily phrased (and solved?) Le Monde mathematical puzzle that does not [really] require an R code:

The five triplets A,B,C,D,E are such that

$A_1=B_2+C_2+D_2+E_2\,,\ B_1=A_2+C_2+D_2+E_2\,,...$

and

$A_2=B_3+C_3+D_3+E_3\,,\ B_2=A_3+C_3+D_3+E_3\,,...$

Given that

$A_1=193\,,\ B_1=175\,, C_1=185\,, D_1=187\,, E_1<175$

find the five triplets.

Adding up both sets of equations shows everything solely depends upon E₁… So running an R code that checks for all possible values of E₁ is a brute-force solution. However, one must first find what to check. Given that the sums of the triplets are of the form (16s,4s,s), the possible choices for E₁ are necessarily restricted to

[sourcecode language=”r” gutter=”false”]
> S0=193+187+185+175
> ceiling(S0/16)
[1] 47
> floor((S0+175)/16)
[1] 57
> (47:57)*16-S0 #E1=S1-S0
[1] 12 28 44 60 76 92 108 124 140 156 172
[/sourcecode]

The first two values correspond to a second sum S₂ equal to 188 and 192, respectively, which is incompatible with A₁ being 193. Furthermore, the corresponding values for E₂ and E₃ are then given by

[sourcecode language=”r” gutter=”false”]
> S2==(49:57)*4
> E1=(49:57)*16-S0
> E2=S2-E1
> S3=S2/4
> S3-E2
[1] -103 -90 -77 -64 -51 -38 -25 -12 1
[/sourcecode]

which excludes all values but E₁=172. No brute-force in the end…

This entry was posted on November 30, 2013 at 12:13 am and is filed under Books, Kids, R with tags Le Monde, mathematical puzzle, R, system of equations. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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