After the Monomatch dice, I was on the hunt for more ways to make a pair of dice that produce the same distribution as taking the sum of two standard dice. I considered using bitwise binary operations, and then tritwise ternary ops, without success. I also tried putting different operations on one die, and then two operands on the faces of the other die. Again, no luck. Then, considering pipped dice, a thought popped into my head: “Try pipwise operations!”
What would that even mean? Well, we could consider each possible position for a pip to be a locus where a binary operation could be performed, with a pip meaning one, and the absence of a pip meaning zero. After performing the operation on all of the positions, we will simply count the ones in the results, just like we count pips in a normal die roll. As for the choice of operation, OR seemed the most natural, as it would be equivalent to just mentally overlaying the faces of the two dice. With a little trial and error, I found this pair of dice:
Notice that the pip positions are based on an underlying 3×5 grid, and that all of the faces have 180° rotational symmetry. The symmetry makes it easier to align a pair of faces so that they are in the correct position to be overlaid. In order for odd rolls to be achievable, both dimensions of the underlying grid must be odd. (Otherwise, there would be no center pip position that maps to itself by rotation, and pips could only occur in pairs.) The number of pip positions must be at least 12 in order to accommodate the highest roll. Under these constraints, 3×5 is the smallest set of dimensions that gives a workable set of pip positions. Another good set of pip positions uses a grid aligned at a 45° diagonal from the edges of the faces, which allows the minimum number of 13 pip positions to be achieved:
It is not hard to come up with a set of faces using this set of pip positions that works.
Another constraint I set myself was that, as much as possible, I wanted to use pip patterns that looked like the familiar ones. There is a danger here that I did not immediately recognize. When I showed the above diagram on Twitter, someone asked, “What happens if you roll a 2 and a 3? Is that 3 or 5?”
My answer, “Yes, depending on which 3 you roll,” was not sufficient. To this person, the different 3’s were obviously the same, because they would be on standard dice. He had difficulty seeing that a 3 oriented as ╱ 90° didn’t just give a pattern equivalent to a 3 oriented as ╲. The problem seems to be that normally when you see a ⚂, you don’t think about the three pips, but immediately jump to the number, for which ⚂ is just an abstract representation, as much as the numeral 3 is. But here a ⚂ is not just a 3! Here’s the full table of results:
You can count the number of faces for each number of pips in the result to verify that it does indeed give the same distribution as 2d6: one 2, two 3’s, three 4’s, and so on.
In related news, I recently received the laser engraver that I backed a Kickstarter campaign for a year ago, so I plan to make some of the dice designs that I’ve written about, as soon as it gets warm enough that I can operate it outside. Meanwhile, I’ve been assembling shaker dice to sell at Gathering for Gardner 14, which will be this April if it doesn’t get postponed due to Covid for a fourth time.
