You have two spheres of different size — say, an apple and a grapefruit. (A proton and a planet will do.) You are going to core each of these fruits. The shape that remains from each sphere is what we will call the napkin ring.
You deftly carve these items such that each resulting ring has the same height (Figure 1).
I claim that the apple ring and the grapefruit ring have the same volume.
Suppose we have a product of functions, uvw. Each component — u, v and w — changes with some additional variable, time, t. How will an incremental increase in time, dt, affect our product?
Imagine your original function, uvw, as a u × v × w box:
“You’ll have to leave a small deposit,” said the shopkeeper. “For security.”
I had just scribbled my mark at the foot of a three-page document, set in 6-point Courier Bold.
I set aside the stylus and met the shopkeeper’s eye. I must have looked worried. He smiled. “A credit card imprint will be fine.”
“Oh. Yeah. Can do.”
I don’t know what I had thought he meant by “deposit.” My left pinky? Something less mundane.
“Have you operated one of these before?” he asked, writing down my MasterCard number next to my signature.
“Umm… No.” It hadn’t occurred to me…
The University of Toronto has posted this page and a half of brain teasers. How many can you solve?
Here’s the first:
Show that n⁷ − n is divisible by 42 for every positive integer n.
First, factor the polynomial.
I once designed a video game for a friend of mine. It featured him running back and forth, Donkey-Kong style, striking objects with the smoke from his pipe. My friend loved the game. But he missed the Easter Egg.
I had included with the game some awkwardly-written documentation. It was headed, “One angry jolt.” Surely my clever friend would read that as “a cross tic” and discover the code in the initial letters of the documentation. He would then read the instructions to hold down the <Ctrl> key while he keyed in the appropriate combination of letters and numbers.