More intransitive dice: Machining precision mathematical toys

A few weeks ago, I posted A Mathematical Valentine's Day gift: Intransitive Dice about machining dodecahedral intransitive dice. But there were at least two problems:

  • I have no reason to believe that the dice I made are sufficiently fair for their intransitive character to be apparent in actual use.
  • Even if they are actually fair, their win ratio is so close that you’d have to make hundreds of throws to have any confidence you would see it in practice.

This means that they are really just decorative objects.

But when my wife first asked whether I could make her some intransitive dice, she had six-sided dice in mind. I was just absorbed by the machining challenge of the dodecahedron, and thought that there wouldn’t be a meaningful challenge making normal six-sided dice with pips to mark the numbers.

Reader, I was wrong.

This seemed like a simple project.

  1. Square three pieces of stock into exact cubes
  2. Plunge 1mm deep holes with a 2mm end mill at 3.25mm spacing to create pips showing face values
  3. Use a roundover bit to round the edges to make the dice more comfortable and help them roll more (be more “live”) when rolled.

Easy, right?

Even cutting them in a careful setup with a fly cutter locked to the same height, I discovered that they varied in thickness from one side to the opposite by up to .04mm. I finally worked out late in the project that brass dust sticks everywhere and is practically invisible, and it’s easy to occasionally miss a tiny piece even when trying to clean thoroughly. So I’m pretty sure that’s the primary reason for the variance.

I used sharpie to mark my cutting plan on the three dice, so that I could keep track of which faces were which and where the pips would need to be cut.

I set up a stop on my vise and used only three table positions to cut all the pips. I found the center, cut all the centered pips, then moved X to -3.25mm and cut all the edge pips, then moved Y to -3.25mm and cut all the corner pips. Some of the pips are not perfectly aligned, and I believe this was because I missed cleaning some chips that got in between that die and the stop at one step. I guess it makes these look hand made. Accurate!

Being not quite perfectly cubes means that the roundover bit is visibly differently asymmetric on some cuts unless I were to adjust it individually for each cut. My solution was to cut each of the 36 edges four times:

  • Conventional cut
  • Climbing spring pass
  • Rotate the part 180° to present the same edge to the cutter but swapping which face was against the vise jaw and which was facing up
  • Conventional cut
  • Climbing spring pass

This made each individual roundover close to fair between its two faces, I think. Mostly. Every time I repositioned a die for its next cut, I swept chips away; I think maybe three quarters of the time I spent rounding over the edges was cleaning between passes. However, there were a few places where the roundover cut in more at one end than the other; some speck of brass dust clearly lifted one end.

Even after rounding them over, the corners were still sharp. I very lightly sanded each face the same amount with 1200 grit sandpaper to reduce tool marks, and then sanded each of the corners the same amount to soften them.

Once I was done with the machining, I used 1ml syringes with fine blunt needles to apply Testors enamel to the pips. I used up about .01ml total (including waste) of enamel in each of black, blue, and green to color the pips and make them easier to read. There are 30 pips on each die, and about 2/3 of the enamel I did use was waste, so there’s about .0001 ml — .1µl — of enamel in each pip! The viscosity of the three colors was substantially different, which kept me on my toes.

I started by roughing out four cubes initially in case I scrapped one. Turns out I scrapped it even before I was finished cutting them down to cubes. I used the lightly-damaged cube to make a more standard 1-6 D6 as a test article, and used what I learned from making it to help improve the intransitive dice I was really trying to make. (I made it different so it wouldn’t accidentally be confused with an item in the set.)

Making those four items took me somewhere between six and eight hours in the shop, not counting all the hours I spent looking things up when I discovered fine details I needed to learn, and also not including making the fine depth stop for my mill that allowed me to plunge the pips to the right depth easily, or the saw arbor for using the slitting saw to cut the brass down to size without making as many wasteful chips. (Those will be their own threads when they are fully complete.)

I’ll post more build details and pictures in comments, but first here’s a family photo of all the dice I’ve made together:


I had a couple fediverse threads of the making.

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Here’s why I made my own slitting saw arbor: instead of making the top of that brass offcut into chips, I sliced it off so that I can may be use it later. I’m such a stock miser!

But then I was scared about whether I was going to experience Rapid Unplanned Disassembly of my saw, so I ran it with a shock absorbing mat in place just in case:

I needn’t have worried. It worked fine. The saw ran true enough that I had both entry and exit tooth lines on the face, and it measured true to my reference side.

Then I sliced that into four raw pieces:

Apparently I took no pictures of truing them up, but I did that with a fly cutter with polished inserts intended for aluminum. The problem I had there is that I initially intended to gang-cut them, but it turned out that they were inconsistent enough in thickness that the vise didn’t hold one solidly, and it moved and scuffed the surface. That’s the die that I ended up practicing on, making it a normal-ish D6.

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While I was working on this, I couldn’t find my 2mm end mill and tried hand-grinding a broken 5/64" drill bit to cut the pips, since that’s basically 2mm. It made 2.5mm pips and wandered:

That test coupon was the offcut from the end after I’d sliced off four cubes. Waste not, want not!

Then I tried a #2 center drill (5/64" tip) which worked but left a burr:

I used that to make the D6, and I spun a larger drill bit in my hand to de-burr the edges of the holes because they were too shallow to use a 45° chamfer bit.

Then I found the 2mm end mill (still in the 4mm collet because it has a 4mm shank), and re-cut the D6 before painting it. It looked a lot better. that way.

I haven’t made a proper vise stop yet — though I have finally decided what design I want to use — so I just superglued a piece of aluminum scrap to the fixed jaw to use as a stop for locating the cubes as I cut all the pips to turn the cubes into dice:


After the roundover bit I ordered arrived, but before cut the real dice today, Joe Pie published a video that included using a roundover bit!

So I used sharpie to mark the faces for the first edge I cut. Here’s a picture partway through rounding over the edges. What looks like a line on the edge of the shapie mark between the three and seven faces on the marked die here is real:

That cut was about .02mm too deep. I blended that out with 1200 grit sandpaper. You can also see the worst-offset pip on the seven face near the bottom.

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This drop of paint barely peeking out the end of the blunt needle was too big and I had to clean up overflow afterward.

Here’s the filled hole:

Incidentally, you can see that even with 3000 RPM on the roundover bit and cranking the table slowly through both passes from both sides, there are still visible milling marks on the roundover. Oh well. You can also see how I knocked down the corners a little.

For the green, I switched to a smaller, bent needle:

This was an improvement.

As the enamel dried, it got darker. In the end, this is going to make it harder to distinguish the dice. If I had it to do over again, I would mix up a few ml with a drop of white paint to make them lighter. It’s possible that I’ll use these as a base coat and then put lighter enamel on top some day, if it makes it easier for my wife to use when demonstrating this.


We couldn’t wait any longer for the enamel to cure. It’s inside the pips; nothing is going to dig it out… We had to give it a shot.

After dinner, we sat down with paper and pencil. We each threw each side of each pair 45 times, and recorded the results.

Five out of the six rounds produced the theoretically expected results. The expected results were A beats B, B beats C, and C beats A. Other than a single round in which A beat C, all the other five rounds ended up with the expected results.

We did have runs of the weaker die beating the stronger die multiple times in a row. But the more throws we made, the more consistently we saw the expected pattern.

Probability. It Works.

Also, with the rounded corners, they were sufficiently “live”. But these being made of brass, we didn’t do this on our nice dining room table. :grinning: