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.

9⁵ + 2⁵ + 7⁵ + 2⁵ + 7⁵ = 92727

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

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).

9⁴ + 4⁴ + 7⁴ + 4⁴ = 9474