Competition problem — Combinatorics

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Combinatorics Difficulty 4.4 AIME Find the answer Italy

Problem:

So-poko is a new puzzle game played on a square board with side $203$ squares. The squares are colored white and black in alternating concentric frames; the outermost frame is black, while the central square is white (see the $7 \times 7$ example alongside). What is the difference between the number of black squares and the number of white squares present in the pattern?

Pick one

Solution

Solution:

The answer is $(\mathbf{E})$ . Let us consider how the difference between black and white squares $d_{k}$ increases in a so-poko of side $k$ . For $k=3$ , the difference is $7$ . Each time two concentric frames are added on the outside (an inner white one and an outer black one), $d_{k}$ increases by $8$ and $k$ increases by $4$ . To go from a so-poko of side $3$ to side $203$ we must add $50$ pairs of frames, so $d_{203}=d_{3}+50 \cdot 8=407$ .

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.