Cover a square with rectangles

Suppose that you have an infinite suply of $m \times 2m$ rectangles for every natural $m \ge 1$ . You have to exactly cover an $\ell \times \ell$ square (you can choose $\ell$ ) by placing $n$ of those rectangles horizontally or vertically. For instance, these are some ways for $n = 2$ , $n = 5$ and $n = 6$ :

(10,6)

(0,0)(0,1)(2,1)(2,0) (0,1)(0,2)(2,2)(2,1)

(4,0)(4,4)(6,4)(6,0) (6,0)(6,2)(7,2)(7,0) (7,0)(7,2)(8,2)(8,0) (6,3)(6,4)(8,4)(8,3) (6,2)(6,3)(8,3)(8,2)

(10,0)(10,4)(12,4)(12,0) (10,4)(10,6)(14,6)(14,4) (14,2)(14,6)(16,6)(16,2) (12,0)(12,2)(16,2)(16,0) (12,2)(12,3)(14,3)(14,2) (12,3)(12,4)(14,4)(14,3)

There are only a few $n$ for which it is impossible to cover a square with $n$ such rectangles. In particular, it is always possible when $n \% 3 = 2$ . Can you prove it?

Input

Input consists of several $n$ such that $n \equiv 2 \pmod{3}$ . Assume $2 \le n \le 62$ .

Output

For every $n$ , first print a line with an $\ell$ between 2 and 30. Afterwards, print $\ell$ lines with $\ell$ characters each. Use different digits, lowercase letters and uppercase letters to indicate each rentangle. Since there are multiple possible solutions, print any one.

Problem information

Author: Salvador Roura

Generation: 2026-01-25T11:38:38.744Z