How Emoji Teleport works
The site never sees your finger. It only knows the color of the square under it, and that is enough.
A hidden checkerboard
Color the grid like a checkerboard. Every animal sits on a dark square and every food on a light one. A step up, down, left or right always lands on the other color. So after an odd number of steps you are on the other color. After an even number, you are back on the same one.
Everyone lands on a food
Animals move 3 steps, an odd number, so they land on a food. Foods move 4, an even number, so they stay on a food. Wherever you started, you are now on a food square.
Taking away the rest
From then on the site knows your color after every move. Before each move it takes away emoji you cannot be on, never yours, and always leaves you somewhere to step. The gaps it leaves steer every possible path to the same square. When only one square is possible, it names the emoji on it.
Do it for a friend
Draw a 4 by 4 checkerboard of animals and foods on paper. Give the same first move. Before each move after that, cross out a few squares your friend can’t be on. Always leave them a square to step to. When one square is left, that is theirs.