New Le Monde puzzle

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When I first read Le Monde puzzle this weekend, I though it was even less exciting than the previous one: find


such that

N(Y-N) is a multiple of Y.

The solution is obtained by brute-force checking through an R program:

Y=2012, N=1006

and then the a next solution is Y=2016, N=504 (with several values for N).

However, while waiting in the plane to Edinburgh, I thought more about it and found that the problem can be solved with paper and pencil. It goes like this. There exists an integer

c such that N(Y-N)=cY.

Hence, solving the second degree equation,

N = left(Ypmsqrt{Y^2-4cY}right)/2 = Y left(1pmsqrt{1-4c/Y}right)/2

which implies that

2 / left(1pmsqrt{1-4c/Y}right)

is one of the integral factors of Y. If we write

Y=qk with 1pmsqrt{1-4c/Y}= dfrac{2}{k},

we get

dfrac{4c}{Y} = dfrac{4c}{qk} = 1 - left( 1 - dfrac{2}{k}right)^2 = frac{4}{k}left(1-dfrac{1}{k}right)

and thus

c=q-dfrac{q}{k} = q,dfrac{k-1}{k}.

We thus deduce that q/k is an integer, meaning that q=pk and thus that k^2 is an integer factor of Y. This obviously restricts the choice of Y, especially when considering that 4cle Y implies kge 4. Furthermore, the solutions to the second degree equations are then given by

dfrac{Y}{2} left( 1 pm dfrac{k-2}{k}right) = left{ begin{matrix} pk\ pk(k-1)end{matrix}right..

The conclusion is thus that any Y which can be divided by a squared integer larger than or equal to 4^2 provides a solution. Now, if we look at the decomposition of

2010=2times 3times 5times 67,

2011=2011times 1,

2012=2^4times 503,

2013=3times 11times 61,

2014=2times 19times 53,

2015=5times 13times 31,

2016=2^5times 3^2times 7,

we see (without any R programming) that (Y,N)=(2016,504) is the smallest solution (in Y). Y=2016 is the smallest solution with N=504 a possible solution in N (N=168 being another one). Which makes (at least) for a more bearable diversion in the plane…

An approach avoiding the second degree equation is to notice that N(Y-N)=cY implies that N and Y share a largest common divider

which implies


thus that Y^prime divides k (because both other terms are prime with Y^prime). Eliminating dividers common to c and N^prime actually leads to Y^prime=k, hence to the same conclusion as before.

Filed under: Kids, R Tagged: Le Monde, puzzle

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