The Mpemba Effect

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.)

Proof That -1 Is Positive

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.

A Double Rainbow

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

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

Root Cause

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.

Penny Wise

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.

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

Heated Prose

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.

Math Notes

95 + 25 + 75 + 25 + 75 = 92727

Mistaken Identity

Suppose a brave Officer to have been flogged when a boy at school, for robbing an orchard, to have taken a standard from the enemy in his first campaign, and to have been made a General in advanced life: Suppose also, which must be admitted to be possible, that when he took the standard, he was conscious of his having been flogged at school; and that, when made a General, he was conscious of his taking the standard, but had absolutely lost the consciousness of his flogging. These things being supposed, it follows from Mr. Locke’s doctrine, that he who was flogged at school is the same person who took the standard; and that he who took the standard is the same person who was made a General. Whence it follows, if there be any truth in logic, that the General is the same person with him who was flogged at school. But the General’s consciousness does not reach so far back as his flogging; therefore, according to Mr. Locke’s doctrine, he is not the person who was flogged. Therefore the General is, and at the same time is not, the same person with him who was flogged at school.

— Thomas Reid, Essays on the Intellectual Powers of Man, 1785

A Canny Gambler

In 1693, Samuel Pepys wrote to Isaac Newton with this question:

“Which is more likely, to throw at least 1 six with 6 dice, or at least 2 sixes with 12 dice, or at least 3 sixes with 18 dice?”

To Pepys’ surprise, Newton found that the first choice has the highest likelihood. The probabilities are 0.665, 0.619, and 0.597 (rounded to three places).

The Mensa Diet

Finding himself hot and overweight at an Air Force base during World War II, Jerry Salny decided he could shed pounds by drinking scotch and soda. Here’s his reasoning:

• It takes 1 calorie of heat to raise the temperature of 1 gram of water by 1° Celsius.
• A glass holds about 200 cc of scotch, soda, and ice. Its temperature is 0° Celsius.
• As he drinks the scotch and soda, his body must supply enough heat to raise 200 grams to body temperature, or 37°C.
• That’s 200 grams × 37°C, or 7,400 calories.
• “Since all the calorie books show scotch as having 100 calories per ounce, and none at all for the soda, we should be able to drink scotch and soda all day and lose weight like mad.”

“This has been tried,” Salny reported, “and although the experimenter hasn’t lost any weight in the process, he doesn’t worry about it much anymore.”

Why doesn’t it work?