A walk on the keypad
The digits 1 to 9 are laid out like a phone keypad: You press a uniformly random key. Then you move to a uniformly random key directly above, below, left or right of it and press that one, and then do the same once more. You have typed a 3-digit number.
Find the probability that this number is even.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
The number is even when its last digit is even. How does one step change the parity of the digit?
Keys next to each other horizontally differ by 1, vertically by 3.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Every move changes the digit by or , an odd amount, so each move flips the parity. After two moves the last digit has the same parity as the first.
Step 1. Horizontal neighbours differ by 1, vertical neighbours by 3. Both are odd, so each move changes the parity of the digit.
Step 2. After two moves, the last digit has the same parity as the first one.
Step 3. The first key is uniform on 1 to 9, and 4 of the 9 digits are even (2, 4, 6, 8). So
The randomness of the two moves does not matter at all.