9⁹ + 1⁹ + 2⁹ + 9⁹ + 8⁹ + 5⁹ + 1⁹ + 5⁹ + 3⁹ = 912985153

When University College physicist Denis Osborne visited Mkwawa Secondary School in Iringa, Tanzania, in 1963, he little expected the question he got from student Erasto Mpemba:

“If you take two similar containers with equal volumes of water, one at 35°C and the other at 100°C, and put them into a freezer, the one that started at 100°C freezes first. Why?”

The other students derided Mpemba, but he was right — in cooking class he’d noticed that hot ice cream mixes froze more quickly than cold ones.

Osborne confirmed the effect and shared a publication with Mpemba in 1969. What’s behind “the Mpemba effect” is still something of a mystery — it seems to be a combined result of supercooling, convection, evaporation, and the insulating effect of frost. (If you want to conduct your own experiment, start with containers at 35°C and 5°C.)

Let

Clearly S is positive. Now multiply each side by 2:

But that’s just the same as S minus 1.

And if 2S = S – 1, then S = -1.

So -1 is positive.

You can multiply these two numbers by simply jumbling their digits:

Remarkably, the same thing happens when you square their product:

Memorize these facts:

With them you can find any two-digit cube root. For example, what’s the cube root of 12,167?

1. Express the number in six digits (012167). Take the first three digits (012) and compare them to the blue cubes above. Find the largest cube that’s less than your three-digit string, and write down its root. Here, 012 is between 8 and 27, so we write down 2.

2. Match the last digit of the number (7) to the last digit of a blue cube above (here, 27). Write down the root of that number (3).

That’s it. Put the two digits together (23) and that’s your root: 23³ = 12,167.

This works for any perfect cube between 1,000 and 1 million.

A currency curiosity discovered by Lewis Carroll:

Write down any number of pounds not more than 12, any number of shillings under 20, and any number of pence under 12. Under the pounds figure write the number of pence, under the shillings the number of shillings, and under the pence the number of pounds, thus reversing the line.

Subtract. [If you need to make exchanges, 1 pound = 20 shillings = 240 pence.]

Reverse the line again.

Add.

“Answer, 12 pounds 18 shillings 11 pence, whatever numbers may have been selected.”

Marie Curie’s laboratory papers are still so radioactive that they’re kept in lead-lined boxes.

Researchers who consult them must agree to work at their own risk.