Competition problem — Combinatorics

Library / /11 of 94

Combinatorics Difficulty 4.2 AIME Find the answer United States

Problem:

The squares of a chessboard are numbered from left to right and top to bottom (so that the first row reads $1, 2, \ldots, 8$ , the second reads $9, 10, \ldots, 16$ , and so forth). The number $1$ is on a black square. How many black squares contain odd numbers?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

$16$ . The black squares in the $n$ th row contain odd numbers when $n$ is odd and even numbers when $n$ is even; thus there are four rows where the black squares contain odd numbers, and each such row contributes four black squares.

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.