Counting with Robot Hands
Previously, we saw that we could count to 35 using our normal human hands, each of which has 5 fingers. If we instead built a robot, and gave it hands, we could put as many fingers on each hand as we wanted (and as our budget would permit). With more fingers per hand, and perhaps even more than 2 hands, we could count higher than 35. On the flip side, we could make much less expensive hands that only contained a single finger, which would require us to have far more than 2 hands in order to count to any reasonably high number.
Video Lecture
Cool Robot Hands
Suppose we design a robotic hand, and we install 15 fingers on that hand. We then attach two of these hands to a robot that we program to count on its fingers, using the same approach we took when counting to 35 on our fingers previously. We can use the same shorthand notation we used with our human hands, but first we need to decide how to represent the numbers 10 through 15. Our notation breaks down if we use 2-digit numbers in place of a single hand digit, so we need to use a symbol that is only 1 character long.
The easy solution to this problem is to start using letters as extra digits. We keep the first digits as 0 through 9, like usual. However, at 10, we write the letter A (in upper- or lowercase - it doesn’t really matter). We write B for 11, C for 12, D for 13, E for 14, and F for 15. This way, we only have a single character representing each hand, and our notation works correctly.
As hinted previously, the next item we have to change with our notation is the subscript that comes after the right parenthesis. We don’t need to worry about the number of characters in the subscript, since that doesn’t matter for distinguishing which hand is which. Our highest digit is F, or 15, and that represents the highest value we can count using only the robot’s right hand. The subscript is always 1 more than that amount, so the subscript is 16. Our notation is therefore:
(LR)16
We count using these cool robot hands in exactly the same way as we did using our human hands. Starting at zero, the first configuration is (00)16. Counting up to 9 yields (09)16. At 10, we still have fingers left on the cool robotic right hand, so we switch to letters and write (0A)16 or (0a)16. We keep using letters up through 15, which is (0F)16.
Once we run out of fingers on the right cool robot hand, we put all the fingers down on the right hand, and we raise the thumb on the left cool robot hand. This is written compactly as (10)16. From here, we keep counting in the same way as before, all the way up to (FF)16, where we run out of fingers.
Now, how high did we count at (FF)16? We could write out all the possible combinations of fingers, but that will take a long time. Let’s take a shortcut instead. Remember that the little subscript 16 means that each finger on the left cool robot hand is worth 16? We can multiply the number of fingers up on that hand (F, or 15) by 16 to get the total value of the left hand, which is 240. Now we just need to add the value of the fingers on the right hand. Each finger on the right hand is still worth 1, so we multiply the number of extended fingers (again, F, or 15) by 1 to get 15. Adding the two hands together gives us 240 + 15 = 255.
Cheap Robot Hands
Although our cool robot hands let us count all the way up to 255, they’re also complicated and relatively expensive to build. Each finger needs some kind of circuit to extend it. We can get away with using a spring to close the finger, but we’ll need an electromagnet to cause the finger to extend (much like a relay moves). That means that each finger needs a spring, an electromagnetic coil, and two wires to supply the coil. Multiply by 15 fingers, and now we have quite a bit of hardware to cram into one robotic hand.
What happens if we go the other way, cheap out on the materials, and just put 1 finger on each hand? Well, in this case, our digits and 0 and 1, since our right cheap robotic hand can only count to 1. In our shorthand notation, the subscript will be 2, since 2 is 1 more than the highest value we can count on the right hand (1 + 1 = 2, so we just passed first grade!).
(LR)2
Our count with two of these cheap robot hands goes (00)2, (01)2, (10)2, (11)2. With only 1 finger on each hand, we can only count to 3! These cheap robot hands don’t seem terribly useful, do they?
