The optimist, the pessimist and the longest sequence

Question

An optimist and a pessimist look at the same finite list of real numbers. The optimist says: "The sum of any eight consecutive terms is positive." The pessimist says: "The sum of any five consecutive terms is negative."

Can they both be right? If so, what is the longest list for which both statements can hold? Give an example of maximum length and prove nothing longer works.

Accepted Answer

Both can be right, and the longest possible list has 11 terms. Twelve terms are impossible. Suppose the list had 12 terms, a1 to a12. Build a table with 5 rows and 8 columns whose entry in row i (i from 0 to 4) and column j (j from 1 to 8) is a(i + j). Positions used run from 1 + 0 = 1 to 8 + 4 = 12, so every entry exists. - Each row is eight consecutive terms: a(1+i), a(2+i), ..., a(8+i). By the optimist, every row sums to a positive number, so the whole table sums to something positive. - Each column is five consecutive terms: a(j), a(j+1), ..., a(j+4). By the pessimist, every column sums t…

The optimist, the pessimist and the longest sequence

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