Intro: Chess Board With Rice: Exponential Growth
It was probably my math teacher who introduced us in High School to exponential growth by telling us the story of the invention of the game of chess. The king (emperor whatever) who had ordered a new game because of the fact that he was bored by the old games, was so happy that he said the inventor: "Name your reward and you will get it!" The inventor asked for a simple reward. "I would like to have one grain of rice on the first chess square, two on the second, four on the third and so on, doubling the amount of rice every square." The legend says that the King was surprised he didn't ask for gold but was quite content that the inventor asked for so little. But when the court scholars told him there wasn't enough rice in the whole world to fill the chess board he had to admit his loss. After that the King executed the inventor or took him as his advisor. There are more versions to this story but the core is always the same: people underestimate the growth of an exponential function.
The math is very simple: the total amount of rice grains is 2⁰+2¹+2²+2³+...+2⁶³=2⁶⁴-1 = 1+2+4+8+16+32+... = 18,446,744,073,709,551,615 (the 64th Mersenne number). But that is still a number. It looks impressive but how impressive is it exactly? (edited: thanks maewart for the alertness).
I am a maker, I love maths and I am a teacher. A few moths ago I thought back to this story and I wondered, wouldn't be nice to visualise the exponential grow of the number of rice grains on a real chess board? I teach physics and every day I see students who understand the sometimes abstract concepts of math better when they see it back in real world situations. So, I built a chess board with rice grains. With real rice.
Step 1: The Measurements and Calculations
You have to do some measurements and calculations. First we have to find out a few things about rice grains. There are many different kinds of rice. I used our regular "house rice": Surinam rice.
Count 100 rice grains and put them on a accurate scale. Mine read 2.0 grams. That means that one rice grain weighs approximately 0.020 g or 20 mg. Then measure the "density" of the rice. It is not really the density because of the air between the grains. Fill a measuring cup to a volume and weigh the rice. My rice filled up 200 mL with 178 g of rice. That means the "density" is 0.178 kg/0.200 L = 0.89 kg/L or 890 kg m⁻³.
You then have to make some decisions and do some shop research. You have to decide how large you want to make your chess board and how you put the rice grains on your board. I wanted to use the board as a demonstration tool at school so I had a new design requirement: make it robust. The rice grains should be inside a housing but, evidently, they should be visible. So I decided to use transparent plastic tubes. I did a very thorough web search and finally bought tubes with a diameter of 60 mm on the outside and 54 mm on the inside. A little bit larger than a normal chess square but I choose for this because I was able to put more rice in them. Off course you can choose for smaller tubes or even square tubes.
Time to fire up your spreadsheet program. My file is included here. At the top of the Excel sheet you can change the parameters so that it fits your design. If you decide to use other tubes, the height and weight comparisons will be off than of course.
Step 2: Materials, Tools and Costs
- wood (I used 12 plates of plywood, 600x300x5 mm)
- extra wood for the bottom and sides
- polycarbonate tube (I used one with an outer diameter of 60 mm and an inner diameter of 54 mm)
- polycarbonate tube with a smaller diameter to put the height examples on
- plexiglass for making the bottom
- paint to create dark and light squares (I used ecoline and varnish)
- material to make supports
- chess pieces
- different kind of height example miniatures (like the Eiffel Tower, moon, Mars, etc). I bought them on Aliexpress, Ebay, Shapeways (3D printed), and 3D printed the Mount Everest miniature myself
- The tools
- laser cutter
- wood tools (knife, hammer etc)
- paint brushes
- 3D printer (only if you have to 3D print a miniature)
I estimate that this has cost €250. If you want to cut costs, you can skip the miniatures. Then it will be around €100. And a bit more boring.
Step 3: The Design
Make a design for your chess board. I did mine without a computer but for clarity I made this drawings. As you can see in the Excel file, the first 13 tubes are not longer than 5 cm above the board. For these "small tubes" a hole in the board of about 2 x 4,5=9 mm deep, with glue to keep the tube in place will be enough. So for these 13 tubes three sheets of plywood of 4,5 mm will be enough. The first two will be cut to hold the tube, and the third one will serve as a ground plate for these tubes.
When the tubes get longer, the weight (remember, they will be filled with rice) and height will not be enough. So you have to make a construction to hold the tubes in place.
For this reason, I designed a system that has a top and a bottom layer. The tubes go all the way through the first three layers and a few cm beneath this layer another layer is made with five layers that are cut and beneath that layer is the bottom layer that consists of a MDF plate of 560x560x16,5 mm): the next two tubes will rest on those plates. With this you make a sturdy construction for the tubes. If you are a bit worried it will not be enough, you can increase the distance between the layers or use thicker layers. You will need more tube then.
As you might have noticed in the Excel sheet I made some calculations on extra height, the extra space for the larger tubes etc. When you do this too, be sure to check them moer than once. Tubing material is quite expensive and you want to cut the right lengths.
If you have access to a laser cutter with more power you can just use thicker plates. The two top layers can be combined into one then.
It will be a kind of educational tool so you can put some information on the chess board. I decided to engrave the number of rice grains. As a side effect I learned that Excel is only accurate to 15 figures. And that provides a problem from 2⁵⁰. The answer Excel gives is: 1,125,899,906,842,620. I noticed the zero's at the and from 50 to 63, so I decided to let Wolfram Alpha do the work. 2⁵⁰ = 1,125.,899,906,842,624. I continued with the results of Wolfram Alpha of course.
You can put miniatures on the squares with no rice on it (because they would be too high). I'll tell about it later. But for the design you can take into account that you can show how high the tube would be if we would continue. For that I used tubes with smaller diameters (12 and 7 mm). To stabilise those, use the same trick as with the larger ones. Make a hole in the top plate and in the bottom plate. I used the laser cutter for these to.
Step 4: The Build: Cutting the Board
First, when you have access to a laser cutter, make a design in the program of your choice. We use CorelDraw (works great for our students). You have to cut out rings in which the tube will fit. You have to experiment here. If you use the exact dimensions of the tube you wil likely end with a too loose or too tight fit. The dimensions are not always precise and the laser cutter is very precise but cuts away wood. It took four iterations before I was satisfied with the result. Check the outer and the inner fit. You will have to place back the circles that you have cut. In the picture you see my pice of test wood. At the time I did this, I was still considering using smaller tubes with a scaling factor. I dismissed that idea very early in the process.
If you are satisfied with the result make the final design of the board. Count the tubes you have to place (in my case there were 18) and cut out the designs in plates of 600x300 mm plywood with a thickness of 5 mm. The complete chess board will be 600x600 mm so every plate you cut will be hal of the area needed.
- First layer (hole side). A plate with 18 holes. The squares are cut too. It might be nicer not to cut them and have a sturdy top layer but that will be very difficult to paint.
- Second layer (hole side). A plate with 18 holes for the tubes. I accidentally made 4 but I two of the three extras in an unseen place (first two bottom layers) and the last extra I used to make the construction stronger.
- Third top layer and first two bottom layer. Three plates with 5 holes for the tubes. I only made 1 because I already made two extra of the above layer which I could use too.
- First layer (non hole side). One plate with no holes.
- Side Plates. Plate 2 and 3 of the top layer on the "other" side. 4 x (20x560 mm) and 8 x (20x300 mm).
I like to think this is possible without a laser cutter. But you have to be very precise because of the alignment. If you make one with only a drill and sanding paper, I would really like to see that!
After making the first cuts, I decided to make some more cuts for my smaller plastic tubes. They will serve as a model for the normal tube and as a stand for the miniature. The positions of the holes, I could only figure out when I got the miniatures. Use a circular saw to make the bottom plate of MDF. At first I decided to make a second one because I wasn't sure the little holes would hold the quite large mass of the three largest tubes. I couldn't find the right diameter for the circle so I decided to cut some more plates with the laser cutter. I had some difficulties to find the right amount because I couldn't find wood with exactly the same height as I used before. I settled for three extra of 5 mm each.
Step 5: The Build: Painting the Board
In order to make nice fields with straight lines between them, cut out the fields of the two top boards with the laser cutter. You could also try to paint them still attached but in my experience that is impossible. Then paint every other field. I used ecoline (coloured ink) colour: 411 (Burned Sienna). Don't forget to paint at about 16 cut out circles too: they will be inside the tube with rice on it. Not all, but the first say 10 will show the colour of the circle so you have to paint them.
When the wood is dried, put them back in the laser cutter and engrave the number of rice grains in them. You have to decide how to order the numbers. I made a " snake", so, I went from left to right and the next line from right to left.
When you paint and then engrave them you get the nice effect that the numbers in the not painted wood are dark and the numbers in the painted wood are light.
To protect the board, paint all squares with a good varnish. Do not forget to paint a few circles: you have to put them back later to put some rice on it.
Step 6: The Build: Cutting the Tubes
The polycarbonate tubes are not very cheap so you have to be careful with them. First beside you have the right sizes. I went over my Excel sheet dozens of times before I took out my Sharpie and started cutting.
So first you have to measure the length you need in your case. Measure this a few times and set a point with a Sharpie. Now you can "connect the dots" if you like. I used some marking tape to be able to cut straight.
Put the tube under an electrical jigsaw if you have access to one or use a hand fretsaw. Be careful, cut gently. I never managed to get a clean cut in one take. Use a sanding machine or sand it by hand. Then cut of the remaining plastic with a sharpie. Document the lengths of the tubes (I put a paper into it with the size) and put them aside.
All tubes will get a bottom. The large ones a slightly sturdier one.
After that it is time to cut out the lids of the tubes. I think it is necessary to protect the rice and the system and put a lid on the tube; you may decide otherwise. Draw circles of 54 mm width and cut them out in a laser cutter. It is always wise to make a testing sample to see if it fits nicely. Put these lids aside too.
Step 7: The Build: Constructing the Board
Now it is time to construct the board. If you have taken care of the measurements this will not be very difficult.
- First put up the sides (I bought some pinewood (100 mm x 19 mm) and sawed four sides of 560 mm + thickness of the wood (19 mm) of the ground plate. It took me some sanding to get a good fit.) Then connect the bottom plate to the side plates by gluing and using screws. You can also use some nails and cut of the heads. I did this because I was afraid the wood was too thin and the screws would split them.
- Then use spare tube to align the holes in the bottom plates. Leave them there and glue them onto each other. Use a weight to glue the (ever) skewed wood.
- Glue the top layers. Be sure to keep the spare tubes to keep the alignment right! If the plates are skewed (mine were) then nail them down too.
- Then attach the sides. Drill narrow holes for the screws and put in the screws.
- Fill the holes of the screws and other irregularities with liquid wood (or a mixture of wood glue and wood pulp as I did).
- Sand the wood thoroughly.
Step 8: The Build: Finish the Tubes
Now it's time to finish the tubes!
- First make bottoms for the tubes. Cut out two circles per tube: one with a diameter of the outer diameter of the tubes and one with the inner diameter. The more precise the better.
- Put some plastic adhesive on the edge of the bottom of the tube. Put some and the sides of the small circle and on one side of the small circle.
- Then glue the small circle on top of the large one and immediately put the tube on them both.
- Be sure to align them carefully: otherwise they won't fit.
- For the small tubes, that only rest on the top part of the board, put in another laser cut plastic circle (glued of course) and on top of that glue in a painted and varnished wooden circle (see step 5). The height should be one layer above the board as it is. The last field will be put in when the tubes are ready.
- For the large tubes, cut a wooden rod with a diameter that is a little less than the inner diameter of the tubes. Glue a few wooden circles on top of that rod. The rice will be put on top of them. You can also choose to fill the whole tube with rice: no one will see that there is no bottom. I didn't do that, but I think I would, if I would make another one.
After you made all bottoms, fill them with rice.
- For the first eight tubes, I counted the rice grains.
- For stability and for the looks, I glued the grains to the bottom. I only did this for the ones where the rice didn't fill up the whole bottom. For those who do, I measured the heights (they were pretty much the heights I calculated in the Excel sheet) so that I can fill them up when I put the board up again after a rest.
It is nice to see the board coming together!
Step 9: The Build: Putting It All Together
It is time to finish the build. First you have to decide if you want to varnish or paint the "box". Because of the imperfections that I caused (I will be a better wood builder next time ;-) I decided to paint the box but, I think only varnishing would be preferable if you make no mistakes.
- After the paint dries, put in the tubes. Start with the large ones and then put in the small ones.
- First lay down the right square. Make sure it is orientated the right way.
- Put in the tube carefully. Make sure the tube is as down as possible.
- Put some glue between the square and the board (not too much) and press the square to the board. Use weights if you can.
- Do this one by one.
Step 10: The Miniatures
When the chess board is ready, it will be quite impressive. But you still have to do the final touches.
- First put the miniatures on the squares they belong. Be sure they are on the right ones.
- You will probably buy other miniatures than I did, so I will only say: try to make a construction that is sturdy and with which you can show how high the tube would be if the you would have continued to make higher tubes. I used smaller tubes as you can see in the pictures and the video to make supports. I am aware of the fact that the diameter of the tubes isn't scaled alongside the height. They would be too thin.
- Put some chess pieces on the side of the board. It might not be recognisable as a chess board without them.
- If you put the chess board on permanent display, you may want to glue/screw everything in place.
- For some miniatures you have to make a dedicated stand. I made one for ISS and the moon, I used a 3D printer and Tinkercad.
- You can also make some miniatures yourself. I made the sun. I really wanted the sun on my board and well, a 3D print is kind of weird. So I took a ping pong ball, put some white LEDs in it, painted it yellow with a marker (I know the sun is white but it looked nicer this way) and made a battery holder and a switch on the bottom. A project ons its own.
- In the smaller tubes I put rice that has been grinded.
- I bought also a scale model of Mars. During the build I decided not to put it on the board. Because of the orbits of both Mars and the Earth around the sun, the distance between them is quite variable. In 2003 the distance was at its shortest, only 56 million km. The farthest distance possible is 400 million km, when Earth and Mars are at opposite sides of the sun. I thought it would be nice to fill in a gap with Mars but it would take too much explaining.
Step 11: Enjoy Your Work
Don't forget to show it to the mathematicians you know.