3D Printed Dice for the Royal Game of Ur

Recently I discovered a cool ancient board game called The Royal Game of Ur. If you’ve never seen or heard of it, definitely look it up. It’s an ancient Summerian Game, a precursor to backgammon that was played for thousands of years. Check out a video entitled “Tom Scott vs Irving Finkel: The Royal Game of Ur”. It’s a very entertaining video telling you all about it and how to play.

You can buy versions of the game but being a maker I wanted to make my own. There are lots of Instructables and plans on Thingiverse and other maker sources for creating your own set. (I’ve got a nice laser cut one I did on Thingiverse… hint, hint shameless plug). The one aspect of the game I never liked was the dice the game uses. Four pyramid binary D4 dice are most commonly associated with game play. I can make the rest of the game, but had to buy and modify D4 dice to properly play.

This Instructable will walk you through my convoluted journey to creating dice for the Royal Game of Ur.

Step 1: A Little Background, and Defining the Project's Goals

After discovering UR I went through my usual new obsessive tendency to learn all about it and research different versions available. There are a lot of really nice examples of boards people have made and being a maker I really wanted to make a nice set for myself. I’m lucky enough to have access to a laser cutter and soon had a nice board and pieces to play with. Then, I made a more travel friendly version that I found on Thingiverse ( https://www.thingiverse.com/thing:2308211 by @cymon) that I love and always take with me on trips.

The problem was the dice. The game uses 4 binary dice and I eventually found blank D4 dice I added the binary tips to. They work great, but I have a problem picking up the pointy pyramid dice to roll. I wanted to figure out how to make a better dice option to play with. For some strange reason I thought of a dreidel and how a small, single die would solve a lot of my problems. It would also be easy for me to make with a 3D printer.

I soon had a working version of a 5 sided spinning dice. I was rather proud of myself and moved on to other pressing projects. What I didn’t realize is I never actually played a game with the die. I assumed it was just what I needed. Later while working on my 3D Printed Percentile Dice Instructable ( https://www.instructables.com/id/3D-Printed-Spinning-Percentile-Dice/ ) a comment from redbeardtrev enlightened me to a major flaw in my project.

As he explained, there was a reason the ancient game used the dice format it did. The use of 4 binary dice didn’t provide random rolls for the game. They skew the odds to favor certain rolls which greatly affect the strategy of the game. I got all caught up in designing a cool, simple die instead of the right dice for the game.

I redefined my design criteria based on how 4 - D4 dice perform, and discussions with redbeardtrev to emulate the proper roll and came up with three options for dice to consider. Number 1 would be a D16 spinning dice with all the proportioned rolls laid out on it. Number 2 would be my original D5 spinning dice but with weighted sides to simulate the proper roll results. And finally for the third option, I would make four stick shaped dice which historically were used around the same time frame in other games. All three options I could design in CAD and 3D print making them easy for me to construct.

Step 2: Option 1 - D16 - 16 Sided Spinning Die

The first option I created was the D16 spinning dice. As with the other spinning dice I made I started in Fusion360 with a 16-sided polygon. Extrude it out, then chamfer the sides to a point. It should look like a flat faced diamond. I didn’t take the design to a sharp point, but filleted the end to give it a slightly round bottom. It spins better, doesn’t mare your table surface as much and transitions smoother as the spin breaks and the faceted sides begin to tumble. From my research in printing spinning dice, the die needs to print flat upside down to be most symmetrical in regards to infill for the print. Because of the shape you can print it hollow, but I wanted a bit of weight to it for feel and to maybe help with the tumbling. Since it prints flat, I added the numbers to the side for better printability. Next pop a small hole in the center of the top for a stem and add a stem slightly undersized to fit. Which is glued in place post printing.

For the number layout, I added up the possible roll outcomes from using four D4 binary dice. You have 4 dice with 2 possible values each. Listing them all out (ex. 0000, 1111, 1001, 0110, etc.) gives you 16 over all roll options of one 0, four 1s, six 2s, four 3s and one 4. I took these values and evenly as I could, spaced them around the die.

Step 3: Option 2 - D5 Weighted Spin Die

Option 2 the weighted D5 proved to be a bit more challenging to design. Based on the 16 possible roll outcomes I figured from above, I calculated the “weight” of each to the total. There is a 1 in 16 chance of rolling a 0 or 4, a 4 in 16 (or ¼) chance of rolling a 1 or 3 and a 6 in 16 ( or ⅜) chance of rolling a 2. Since 2 is the highest odds at 6 out of 16, that would be baseline at 100% weight (solid) of a side. By running the proportions for each of the other sides (ex. 100%/X=⅜(value of 2)/1/4(value of 0 or 4) ) and solving for X you would get side 2 = 100%, sides 0 & 4 = 16.66%, and sides 1 & 3 = 66.66%. By using volume of each side, I leave side 2 solid, sides 0 & 4 I take 83.33% (100%-16.66%) of the total away, and sides 1 & 3 I take 33.33% (100%-66.66%) of the total away.

To construct it, I started with a 5 sided polygon, extruded just one wedge or side of it. This would be the #2 solid side. Copy and rotate the wedge then I copied that wedge and scaled it to 33.33%. Center this and cut from the wedge it’s in. This is side 1, copy it and rotate to position to create side 3. Repeat this procedure again but scaling at 83.33% to make sides 0 and 4. Then to balance the die, I distributed them around to even out the balance so it would spin better. Combine all wedges into one body. Next I numbered the wedges on the flat face. Place the numbers opposite the wedge they represent. For a #2 roll, the solid side would land flat, so the number is placed at the point opposite and above. Next I added a shallow hole in the center and a cylinder stem like the D16 spindice. When printing, I added 25% infill to give more weight. In hindsight, The infill should be solid to maximize the weighted effect.

Step 4: Option 3 - 4 Binary Stick Dice

For the stick dice option, I just wanted to basically make long rectangles with two blank and two marked sides. I also wanted to print them vertically so the infill of the print wouldn’t weigh one side or the other and throw off the roll. I drew an equal sided square then added small arcs to each side. I figured if the sticks had soft edges they would tumble better and perhaps add to the randomness of each die’s roll. Extrude this shape to create a long rectangle. I filleted the ends to soften them. Next I added a design with text to signify a positive roll. I just used 4 asterisks wanting a simple pattern that would be enough to show a positive roll but hopefully not influence the weight of the die and therefore throwing off the randomness of the roll. Extrude and cut out the text just deep enough to show. Repeat on the opposite side. Again, I printed these vertically and liked the feel of at least 25% infill. I think now a heavier die will roll and tumble better.

Step 5: Testing - Was It All Worth It

Now the fun part. To test the dice, I rolled each option 100 times and charted the results. I was pleasantly surprised that the outcome of the control D4 dice fit so nicely in my predictions. When you add the roll data from the 3D printed options you can really see what works and what does not. The stick dice are pretty much spot on, as I suspected. These make a great printable, historically accurate option. I was also happy to see the D16 spindice fit so well in the curve. The weighted die proved more elusive to get correct. It was a bit of long shot to think I’d get it right the first time, still a bit disappointing. There wasn't much difference between a weighted and non-weighted version and they didn't come close to matching the D4 control curve. I may revisit the idea later and if anyone is interested in pursuing it, I’ll include the Fusion files to play with.

My main goal was to create a 3D printable compact unique die to use with the Royal Game of Ur sets I make. I think I accomplished this with the D16 spindice and the stick dice, so 2 out 3 ain’t bad!

Step 6: Files and Instructions

I've included all the options here for you to print and experiment with if you like. I also included the Fusion files if you really want to dig deeper and an STL of game pawns to print and use. You will also see two paper prints of the Royal Game of Ur board and rules that I created for a summer camp for kids. The campers got to color their boards then learn to play. We used colored buttons for the board pieces and actually molded D4 dice for them in the camp from a silicone mold and resin I made. The kids really enjoyed that process.

For the dice prints, I would increase the infill as much as possible or desired. I did 25% infill for all of them. I think if they were a bit heavier it would have a better feel. Size them how you like. I liked the stick dice at 60% original. That made them about an inch long and you could shake them well in your hand for a good roll. The D16 was nice at full size, but its about 1.5 inches in diameter. I made a 50% version that's tiny, but will fit in my travel set.

I hope this Instructable was enjoyable, and I really hope you explore the Royal Game of Ur and learn to love and appreciate the history of this enjoyable game as much as I have.

Step 7: How About One More...

Thanks to the suggestion from @redbeardtrev I've added one more dice option. This is a stick die format that requires only two dice. By putting two - 1s, one - 2, and one - 0 on each of the die, you will get the same odds of using the other dice options laid out above. This is an efficient way to get the proper dice for UR. As you can see from the included chart the numbers lay out almost identical to the others. For each, I rolled the set of dice 100 times and logged the results.

I also started to lay out the 16 sided stick dice, but after drawing in Fusion360, it seemed pretty obvious it wouldn't be efficient to roll, so I didn't bother printing one - having much better options.

Hope you enjoy these varied dice for the Royal Game of Ur.