Electronic Design

# What's All This Mental Math Stuff, Anyhow?

Some night when you can't get to sleep, try these puzzles in your head in the dark. They're at least as good as counting sheep. I've done them in my head, and they're kinda fun.

1. Little Egbert bought a mountain. Its shape was a perfect hemisphere with a 5270-ft radius, set on a flat plain. He decided to build a railroad to transport him to the top. It was a monorail, which made it very easy to plan, with minimum width, and only one rail. The rail was offset 10 ft away from the surface of the mountain, to avoid digging, so the radius was exactly 5280 ft. The train could only ascend a 4% grade. The basestation was at the very bottom of the mountain, so the train could not get a running start up the hill. How much track did Little Egbert have to buy? (a) 25.00 miles (b) 25.020 (c) 25.040 (d) 25.40

1b. For extra credit, if the train leaves the basestation going north on the west side of the mountain, from which direction does it approach the summit? I don't have a really solid answer for this one. You tell me!

2. Little Egbert was walking across a field at 2.000 mph. The closest road ran true north, but the place he was going was 2 miles north and 0.1 miles east of his starting point. So he walked straight across the field, the shortest route between two points. His brother Pythagoras decided to walk along the road at 2.000 mph, for 2 miles, and then cut true east. When he got to the corner and started east, he saw that Egbert was quite close to the end point. He decided to speed up to get there at the same time. How fast did he have to walk to achieve that ? (a) 8.0 mph (b) 16 (c) 80 (d) 160.1 (e) 442

3. It's easy to set up two double-pole double-throw switches, one at each end of a room, so people entering the room can turn a common lamp on or off with either switch. Little Egbert added a new kitchen and dining room to his house. The kitchen was triangular with three doors. How could he arrange three switches so that anybody entering at any door could turn the main lights on and off just by throwing the adjacent switch? The dining room had four doors. How could he arrange switches at all four dining room doors with the same capability? (Note, I just got a new kitchen with three doors and a new dining room with four doors.)

4. Here's another sphere problem. Little Egbert sat atop a 16-ft stepladder, in Quito, Ecuador, astride the equator. He lowered a weight by a thread from a well-defined point at the top of the ladder to mark the place on the ground directly below it. Then he pulled up the weight—and waited—and dropped it. It hit the ground at a place not exactly the same as the first point. What was the difference in position? This goes to show that when you drop something, it does not just fall "down." (Unless you are at the North Pole or the South Pole.) Assume the radius of the Earth is 4000.000 miles and that g = 32.0 ft/s2. Thus, the rotational velocity of the Earth is 25,132.74 miles/24 hours, or 1047.20 mph, or 1535.89 ft/s, or 18430.70 in./s. An accuracy of 1% is requested. (If you have a release mechanism that does not affect the weight's transverse motion, you can find out how far you are from the equator. Or, you could find out if you have a release mechanism that does not affect the weight's transverse motion by rotating the release mechanism in various directions.) (a) 1.4 mils (b) 14 (c) 34.2 (d) 112

5. Compute the square root of 156 to a precision better than 1 ppm. (It is easy to compute to 1 ppb.)

The answers will be published in Electronic Design. Complete solutions with explanations will be posted on my Web site at www.national.com/rap on April 15. (I apologize in advance for any complaints that I should have written this in metric terms. Sorry, but that's not going to happen. Brain puzzlers do not start out "There was a hemispherical mountain with a radius of 1609.265 m...")

Comments invited! [email protected] —or: Mail Stop D2597A, National Semiconductor P.O. Box 58090, Santa Clara, CA 95052-8090