Le Monde puzzle [#848]

The X’mas version of Le Monde mathematical puzzles:

A shop sells X’mas trees (with 25 models which prices range from 36€ to 65), X’mas fairy lights (with 18 models which prices range from 6.50 to 27), and X’mas balls (with 13 models which prices range from 9 to 17.50). A lot is made of a model from each group. Is it possible to set the prices so that all different lots have different total prices ? Differences are expressed in cents.

It would be straightforward to set an R code picking 25x18x13=5850 prices at random until all sums are different or a maximal number of trials is reached… but the maximal number of values for the total prices is

100x(65.01-36)+100x(25-6.49)+100x(17.51-9)=5803

so this proves impossible to satisfy. No wonder the title of the puzzle is “A X’mas gift”! (The most interesting entry in this issue of the Science leaflet is an impressive photo of an amputated hand grafted on a leg by Chinese surgeons while waiting for the damaged arm to recover. Maybe too impressive at X’mas dinner time…)

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