Problem 5

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

Step 1 of 5: Assume 17 terms and form the overlapping array

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a_1,\ldots,a_{17}\text{ arranged in }7\text{ rows of }11\text{ consecutive terms}

Detailed analysis

If a sequence had more than 16 terms, take its first $17$ terms. Make seven rows: row $i$ contains $a_i,a_{i+1},\ldots,a_{i+10}$ for $i=1,\ldots,7$ . Each row is an eleven-term block, while each column is a seven-term block.