I noticed a word "code" in a math problem of an entrance examination of Kaisei, a prestigious private junior high school in Tokyo. (I appeared in an ad of a prep school on a newspaper on 6 February 2022.) The following outlines the problem. It is merely about combinations rather than cryptography, as expected.
A code is represented by a 2-by-7 matrix. A filled square cannot be adjacent to another filled square horizontally or vertically. The orientation of the matrix is fixed.
(1) How many squares can be filled at maximum? How many codes can be made with such maximum filling?
(2) Consider the case in which five squares out of the fourteen are filled.
(A) Draw all the codes that can be made without filling the squares in the first and third columns from the left.
(B) How many codes can be made without filling the squares in the third and fifth columns from the left?
(C) When five squares out of the fourteen are filled, how many codes can be made?
(3) Let us consider how many codes can be made as the number of columns used is increased (e.g., only the leftmost column, only the two leftmost columns, ...). The case of no filling is counted as one. For example, the number of patterns to fill the leftmost column is three (as shown in Fig. 2).
(A) When only the two leftmost columns are used, draw all the patterns other than the one with no filled square.
(B) Consider only the three leftmost columns. How many codes can be made?
(C) Consider all the seven columns. How many codes can be made?
KEY: (1) 7; 2; (2)(B) 8; 102; (3)(B) 17; 577
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