Suppose your dog eats the ace of spades and you’re forced to play poker with a 51-card deck. Does this make it more or less likely that you’ll be dealt a two-pair hand?

The answer is not immediately clear. The loss of the ace means that fewer two-pair hands are possible, but the total universe of possible five-card hands is also reduced.

Surprisingly, the probability is unchanged. “[T]he probability of getting a Two-Pair hand in the 51-card deck with the missing Ace of Spades is exactly the same as for a 52-card standard deck,” notes George Mason University mathematician D.M. Anderson. And “[i]t turns out that this seemingly magical result for a deck with a missing Ace of Spades is not limited to Two-Pair hands. Indeed, examining the traditional hierarchy of 5-card poker hands, we find an analogous result holds for One-Pair hands, Three-of-a-Kind, Full Houses and Four-of-a-Kind.”

(D.M. Anderson et al., “Not Playing With a Full Deck?”, Recreational Mathematics Magazine 11:18 [March 2024], 51-63.)

This remarkable phenomenon was discovered by Cambridge mathematician James Grime. Number five six-sided dice as follows:

A: 2, 2, 2, 7, 7, 7
B: 1, 1, 6, 6, 6, 6
C: 0, 5, 5, 5, 5, 5
D: 4, 4, 4, 4, 4, 9
E: 3, 3, 3, 3, 8, 8

Now, on average:

A beats B beats C beats D beats E beats A

and

A beats C beats E beats B beats D beats A.

Interestingly, though, if each die is rolled twice rather than once, then the first of the two chains above remains unchanged except that D now beats C — and the second chain is reversed:

A beats D beats B beats E beats C beats A.

As a result, if each of two opponents chooses one of the five dice, a third opponent can always find a remaining die that beats them both (so long as he’s allowed to choose whether the dice will be rolled once or twice).

(Ward Heilman and Nicholas Pasciuto, “What Nontransitive Dice Exist Among Us?,” Math Horizons 24:4 [April 2017], 14-17.)

From Pi Mu Epsilon Journal, November 1950:

(1/2)³ < (1/2)².

Taking the logarithm to the base 1/2 of each member of the above inequality, we write

3 log_1/2(1/2) < 2 log_1/2(1/2).

But log_bb = 1. Therefore

3 < 2.

In his 1985 commonplace book Michelangelo’s Snowman, George Herrick gives a notable mnemonic for the bones of the wrist:

Some Lovers Try Positions That They Can’t Handle

(scaphoid, lunate, triquetrum, pisiform, trapezium, trapezoid, capitate, hamate)

Two more, from the r/Anatomy subreddit:

According to Frank M. Chapman’s Color Key to North American Birds (1912), the hooded warbler sings You must come to the woods, or you won’t see me.

Ohm’s law states that V = IR, where V is the voltage measured across a conductor, I is the current through the conductor, and R is the conductor’s resistance. In the image mnemonic at left (easily remembered by the word “viral”), covering any of the unknowns gives the formula in terms of the remaining parameters: V = IR, I = V/R, R = V/I.

Wikimedia user CMG Lee has devised other mnemonics in the same style for high-school physics students (right). For example, F = ma, m = F/a, a = F/m. In the corresponding SVG file you can hover over a symbol to see its meaning and formula.