## Le Monde puzzle [#1013]

**A** purely arithmetic Le Monde mathematical puzzle:

An operation þ applies to all pairs of natural integers with the properties

0* þ (*a+1) = (0 *þ a)+1, (a+1) þ (b+1)=(a þ b)+1, 271 þ 287 = 77777, 2018 þ 39 = 2018×39*

Find the smallest integer d>287 such that there exists c<d leading to c þ d = c x d, the smallest integer f>2017 such that 2017 þ f = 2017×40. Is there any know integer f such that f þ 2017 = 40×2017?

The major appeal in this puzzle (where no R programming seems to help!) is that the “data” does not completely defines the operation * þ *! Indeed, when a<b, it is straightforward to deduce that a* þ *b = (0* þ *0)+b, hence solving the first two questions by deriving (0* þ *0)=270×287 [with d=315 and f=2017×40-270×287], but the opposed quantity b* þ *a is not defined, apart from (2018-39)* þ *0. This however brings a resolution since

(2018-39) *þ *0 = 2017×39 and (2018-39+2017) *þ 2*017 = 2017×39+2017 = 2017×40

leading to f=2018-39+2017=3996.

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