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# Involuntary Action

## Solution

author: Daan van de Weem

We are presented with 25 textboxes, initially filled with 1's, and four buttons. If we try playing around with the buttons, we find that they replace the numbers in the boxes with different numbers. We also find that, subject to rounding errors, clicking a button twice in a row restores the numbers to their original values. Taking a clue from the title, we find that each button applies an involution matrix to the numbers (an involution matrix is its own inverse). Thus we also see that the twenty five boxes correspond to a 25-vector.

We can (tediously) determine the value of each matrix by inputting vectors of the form (1,0,0,...), (0,1,0,...) etc for each button, thus yielding the corresponding column of the matrix. A property of involution matrices is that, in their eigenbasis, their diagonals always consist of either +1 or -1. Taking a clue from the matrix dimenions (25x25), we can count the number of +1's in each matrix's diagonal (in its eigenbasis, or alternatively use the trace which is invariant with respect to change of basis). This number ranges between [0,25], and so we can map it to the letters, [A,Z], yielding the answer, `JERK`.