Suppose you wanted to construct a set of Wang tiles that performed base-10 addition. Like, you would start with a row of interlocking puzzle-piece tiles that said “ 3280+834 ”, with a flat edge along the top, left, and right, but a ragged edge along the bottom, and all the possible ways to complete a smooth-edged shape would have the correct answer “4114” on the bottom row.
This involves four kinds of top-row tiles: digits, left boundary, right boundary, and “+”. (Or 13 kinds, if we count each digit separately.) These can mostly all fit together in an arbitrary order, but the left and right boundary have smooth edges that won’t interlock with anything else. And if you put more than one “+” sign in there, you will have created an unsolvable puzzle. (You could use two different kinds of digit tiles, one for the left addend and one for the right, in order to make this impossible.)
The first order of business is to get the corresponding digits lined up, so that you get some kind of representation of “(3+0), (2+8), (8+3), (0+4)”. This involves introducing an extra “0” to the “834”, and then shifting digits over, one per row, until they reach their final position. Then, once the corresponding digits are lined up, you need to reduce the pairs to sum digits, one by one, and finally once everything is a sum digit, you can finish off the bottom with a nice smooth-edged line.
Here we’re copying the original problem down, row by row; so we have “copy digits” whose tops interlock (only!) with the bottoms of the corresponding digits above them, and similarly a “copy left edge” and “copy right edge” and “copy plus sign”. Except that there are 120 kinds of extra “digit insertion tiles” which allow you to, under the right circumstances, shift a number over by one tile to the right. Each of these kinds of digit insertion tiles has a digit printed on it that is determined by its left and bottom edges, which are the same, and a right edge that is the same as its top edge. There are top-edge versions for each of the possible digits that could be above, and also the “+” tile and the right-edge tile.
The “copy down +” and “copy left edge” tiles have a special right edge that allows an “insert leading 0”, so you can add an extra-leading-zero tile to shift over either the left or the right operand by one. The “copy down” tiles are not “shift tiles”, so you can only use them directly below the thing they’re copying, so you can only insert one leading 0 per row.