In this project, you’ll create a randomizer by hammering nails into a board. What do I mean by a randomizer? Well, imagine a lot of nails standing like trees in a forest on a steep slope. If you roll a marble among them, it should bounce around and come out at the end with an equal, random chance of being on the left side or the right side. If you label the left side “yes” and the right side “no,” you can use the randomizer in games, or to make difficult decisions, or just for fun.

You can also use it to demonstrate some trickier aspects of probability, which I’ll describe at the end.

Previously I mentioned that one-by-six pine is actually ¾" x 5½". When you go shopping for it, probably you will be expected to buy a piece 96" long, like the two-by-four that you were using previously. You only need 12" of one-by-six for this project, but you’ll be using more of it later in the book, and it won’t cost very much. I think 96" will be worthwhile.

The hammer can be the cheapest you can find. If you are physically small or very young, you can consider choosing an 8-ounce hammer, bearing in mind that the nails in this project are small and don’t require a lot of force. 10-ounce and 12-ounce hammers may also be found. Figure 4-1 compares an eight-ounce and 16-ounce hammer.

Long-nosed pliers are the easiest type to use in this project, as shown in Figure 4-2. If you only have generic slip-joint pliers like those in Figure 4-3, you can use them instead.

To learn what a finishing nail is, and a lot more facts about nails, see the Hammer and Nail Fact Sheet on page 48. A box of 1¼" finishing nails will probably look like the one in Figure 4-4.

For the marbles, if you don’t have any, you can buy them super-cheaply from Walmart or eBay. The size must be ⁹/₁₆" diameter (or 14mm), to match the spacing between the nails.

Safety While Hammering

Before you start hammering nails, consider whether you should be wearing eye protection. There is the very slight risk that when you hit a large nail, a metal fragment could break loose and fly off at dangerous speed. Personally I have never seen this happen, and in any case, the projects in this book only require small nails. I’m just mentioning the safety risk for future reference.

Hitting your thumb is a more realistic risk, because everyone does that sooner or later. You can eliminate the risk by using pliers to hold each nail, but after a while you may get tired of doing this.

Hammering Practice

This project requires accurate nail spacing, so it’s a good idea to get in a little practice. You can use a scrap piece of two-by-four to test your hammering skills. First check Figure 4-5 to make sure you have the basic concept figured out.

Starting the nail is easy. With your left hand (if you are right-handed), use the pliers to hold the point of the nail against the location where it’s going to go. Hold the nail vertically, hold the hammer near its head (although not so close that you can bang your fingers against the nail), and give the nail two or three small, gentle taps.

When the nail is able to stand up straight on its own, take the pliers away. Shift your grip toward the end of the handle of the hammer, and whack the nail a few times, being careful to bring the head of the hammer straight down. Holding the end of the handle allows you to apply greater force, because the hammer is like an extension of your arm, multiplying its leverage.

However, when you are using it this way, the hammer becomes more difficult to control. If you find that you aren’t hitting the nails on-target, shift your grip closer to the head of the hammer. Your strokes will be less powerful, but you can make them more accurate.

What if a nail starts to go in at an angle? Straighten it immediately with your pliers. And if a nail gets bent? Pull it out with the pliers, and throw it away.

Nail History

Long ago, in colonial America, nails used to be made by hand. They were so precious that if a house was abandoned, people would burn it down just to retrieve the nails. Thomas Jefferson derived some income by owning a blacksmith business that made and sold nails.

These days, nails are so cheap, we don’t need to waste time trying to straighten them.

A claw hammer is designed so that you can use it to pull out a nail, as shown in Figure 4-6. In this project, the nails will be spaced too closely to allow this, and pliers are the only option. The nails are so small, you should be able to pull them out without much trouble.

Splitting Wood

If you hammer a nail too close to the end of a piece of wood, the force of the nail can break the wood apart. The risk increases if the nail is too thick, or too close to the end of a piece of wood. Hardwood may split more easily than softwood. Thin wood splits more easily than thick wood. And if you insert two or more nails in line with one line of the grain, the wood is more likely to split.

My successful attempt to split a piece of pine is shown in Figure 4-7. I had to use a relatively thick nail to do it. I had a much easier time splitting particle board, as shown in Figure 4-8.

It’s a good idea to see if you can split a piece of scrap wood, just so that you learn the limits. Using the little 1¼" nails that I specified for this project, I doubt you will be able to split a piece of pine—but you can probably split some ¾" square dowel, especially if it is hardwood, which is less willing to make room for the nail.

Cutting the One-by-Six

You will need a 12" section of one-by-six pine that has no knots in it. Knots are unacceptable for this project, because they are hard and brittle, so you won’t be able to hammer nails into them.

How should you go about cutting your section of one-by-six? Bear in mind a one-by-six is too wide to fit in your miter box. If you had a panel saw and a sawhorse, that would probably be the most obvious way, but I am assuming that you are working on a table top with a tenon saw.

With this in mind, Figure 4-9 shows my suggestion. You will draw a guide line across the board with your speed square. Underneath the board you will put a sacrificial two-by-four which is 5½" long—the same length as the width of the board. (You can cut the sacrificial piece in your miter box.)

Place the sacrificial piece so that it extends out over the edge of your work area, as shown in the figure. This will allow room for two clamps, and for you to make a saw cut. The figure shows a cut in progress, although I took the saw out of the photograph so that you can see the setup more easily.

Notice another spare piece of two-by-four that I tucked under rear end of the board, just to keep it level.

If you feel you need a guide piece to keep your cut straight and vertical, as in the previous project, you could add another 5½" section of two-by-four for this purpose on top of the board, and extend the clamps upward to hold it. But the precision of this cut is not crucial, and I think you should try making it without a guide piece, just to test your sawing skill.

You don’t even need the sacrificial piece, if you don’t care about splinters. But that’s up to you.

When you have your 12" of knot-free one-by-six board, you need to make sure it is smooth, so that marbles will roll around on it freely. Sand the board until you cannot feel any ridges of the grain when you run your fingers over the surface.

Making a Plan

For this project, the array of nails has to be in a triangular pattern, but I figured out a simple way to draw it from a grid of rectangles. All you need is a letter-sized sheet of paper, a ruler, and preferably some pens in three colors.

Begin with the grid shown in Figure 4-10. I’ve shown the dimensions in millimeters, because they are a lot easier to deal with than sixteenths of an inch. Inches are mandatory in the workshop environment in the United States, but there’s no need to make things more difficult than necessary when you’re just working with a pen and paper.

I’ve drawn the rectangles in green, because I’m going to add other colors.

Now using black ink, draw the diagonal lines shown in Figure 4-11, connecting the corners of the rectangles. Try to make the lines accurate.

Make red dots where the black lines cross, as shown in Figure 4-12. These will be the positions where you hammer in the nails. Their positions are important, because the marbles will roll among them. Extra nails will be inserted on the black dots, and their positions are not so important, because their purpose is just to stop the marbles from rolling out and escaping from the grid.

How will you transfer this pattern onto the wood? I’ll show you as soon as I deal with some necessary setup.

Precision Hammering

One of the most important factors, when you are hammering nails, is to have a firm support. If you rest the board in the center of a table, the table will tend to vibrate. This means that a lot of your energy will be wasted, causing the table to jump around.

The best support would be a very solid workbench, but if you only have a table, position your work over a leg of the table, to transmit your hammering force directly to the floor. I am assuming that you are using a wooden-topped table, not a glass-topped table!

Sit close to the work. If you were hammering 3" nails, you might want to stand back and apply some body weight to the hammer. For this little job, that isn’t necessary. Position yourself close to the nails so that you can see them clearly.

Tape your plan to the piece of two-by-six, and use an awl before you start hammering. This sounds like an unnecessary extra step, but it will save you time. Position the sharp point of the awl exactly where you want each nail to go. Make sure the awl is vertical, and push down through the paper to make a prick mark in the wood. Now when you hammer in the nail, it will position itself automatically on the mark, and your job will be much easier. When I built this project, I began by making prick marks with the awl for all of the nail locations.

Hammer the nails through the paper into the wood, as shown in Figure 4-13. When you finish hammering, you can rip the paper away. This eliminates the need to draw lines on the wood and erase them later.

Depth Control

If you hammer in a nail too far, it will go all the way through the board and emerge underneath. To avoid this, you need to do some measuring.

Bearing in mind that the board is ¾" thick, and each nail is 1¼" long, how deep should you hammer each nail?

It will be easier to figure this out if you convert everything to eighths of an inch. The ¾" board is ⁶/₈" thick. Each nail is ⁵/₄" which is ¹⁰/₈" long.

To make sure the nails don’t go all the way through, you can hammer them ⁵/₈" into the wood. That will leave a ¹/₈" margin of error.

Because each nail is ¹⁰/₈", the exposed part will also be ⁵/₈". When you start hammering, you can use a ruler to check how much of each nail is sticking out, and Figure 4-14 shows me doing this. In this figure, you can see that ¾" of the nail is exposed. But ¾" is the same as ⁶/₈". So I can hammer it in another ¹/₈".

In Figure 4-15 I added a second nail at the other end of the diagonal line where the first nail is located, and stretched a piece of yarn between them. Now the nails along that line can be hammered in so that they are all at the same height as the yarn.

Stretching a string is a method often used on construction sites—for example, to make sure each layer of bricks in a wall is properly aligned. The first row of nails is shown in Figure 4-16.

Some people enjoy being precise. Others don’t. If you don’t, you can skip the figuring, and complete this project just by guessing the depth of each nail and laying your ruler along them once in a while to verify that they are about right. You will run a greater risk of a nail going all the way through and poking out at the bottom, but maybe that won’t be important to you. The main thing is for you to complete the project in a way that satisfies you.

Rolling Randomly

When your nailing work is complete, it should look like Figure 4-17. There are about 120 nails; how long do you think you will take to hammer them all in? Suppose it takes you 15 seconds per nail. That would add up to about 30 minutes. But I think you’ll get quicker as you proceed. The project is not as challenging as it looks (so long as you use the awl to locate each nail properly).

Now you can rip the paper away. Tilt the wood upward at a 45-degree angle, and drop a marble in at the top. Can you predict where it’s going to end up, as it bounces among the nails? I think not. That’s why this little gadget is called a randomizer: it is unpredictable. Try dumping several marbles at once, and see where they go, as in Figure 4-18.

One thing you’ll notice is that some marbles get stuck at the edges of the board. In many of the projects in this book, I’m going to leave you room to make improvements. How would you prevent the marbles from lodging like this? Could you add a few extra nails to take care of it?

Here’s another question. What if you try to cheat by dropping a marble a little to one side, between the topmost nails? This might affect the outcome, so maybe you should take a precaution against it. You could add a couple more nails above the ones at the top, and space them the precise width of a marble. Now it should fall exactly the same way each time—or will it?

I think you will find that tiny variations are still enough to have an effect. The marble may have a little spin on it, or there may be a current of air, or some other factor. In science, factors of this kind are known as uncontrolled variables.

Suppose the left area at the bottom is labeled “yes” and the right area is labeled “no.” If this is a really random device, and you test it many times, you should expect a marble to end up an equal number of times in the “yes” and “no” areas. You can try to verify this, perhaps using multiple marbles to speed things up. If the outcome is equally likely to be “yes” or “no,” we say that the options are evenly weighted.

Do you think you would have affected the outcome if you spaced the nails farther apart? The marbles might bounce around more, making their exit point less predictable. Or would they?

What if you tilt the wood more or less? Will that affect the randomness of the outcome?

The Bean Machine

I designed the randomizer as a smaller version of a device sometimes known as a bean machine. An example is shown in Figure 4-19. It contains dozens or hundreds of small, hard objects, such as beans, which can be dumped into a reservoir at the top. Imagine that a pane of glass, in the front, prevents the beans from falling out. When they trickle down among the pins, they stack up in a heap shaped like the one in the figure. More beans accumulate in the middle than at the ends because there are more possible paths that the beans can take to the middle.

This is an important concept. If you look at the extreme right-hand end of the reservoir at the bottom, there is only one path a bean can take to end up there. But by my count, there are 17 possible paths leading to the next position along, and there are probably hundreds of different paths leading down through the nails to the center positions.

Incidentally, “bean machine” is a widely used name for this kind of gadget. It’s not just something that I made up. It even has a Wikipedia entry. You can check it out.

The profile of the stacked beans at the bottom is often known as a bell curve, because it is vaguely shaped like a bell. This curve crops up in many areas of statistics. SAT scores, for instance, tend to be distributed in a bell curve, with most people scoring near the middle and only a few at the extremes. Other examples are inaccuracies in production processes, variations in value of a blue-chip stock over a long period, and some astronomical phenomena.

The formal name for a bell curve is normal distribution, which is a big concept in the social sciences, and also in mathematics. If you had enough patience, you could make a bean machine to model normal distribution fairly accurately. But why does it turn up in so many different places? That’s something you’ll have to read about on your own. Meanwhile, have fun testing the randomness of your randomizer.