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We consider the division of a chess board

8 \times 8

in p disjoint rectangles which satisfy the conditions:
a) every rectangle is formed from a number of full squares (not partial) from the 64 and the number of white squares is equal to the number of black squares.
b) the numbers

\ a_{1}, \ldots, a_{p}

of white squares from

p

rectangles satisfy

a_1, , \ldots, a_p.

Find the greatest value of

p

for which there exists such a division and then for that value of

p,

all the sequences

a_{1}, \ldots, a_{p}

for which we can have such a division.
[color=#008000]Moderator says: see https://artofproblemsolving.com/community/c6h58591

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Problems(1)