The Cable Guy Paradox

The cable guy is coming tomorrow between 8 a.m. and 4 p.m. Let’s bet on whether he turns up in the morning or the afternoon.

Both windows are four hours long, so as we sit here today, it seems rational to treat them as equally likely. But suppose you choose the morning. As the clock begins to tick, the morning window will gradually close, making the afternoon seem increasingly preferable. Though your present self regards the two eventualities as equally likely, it seems that your future self won’t. Should that affect your decision today?

(In my experience the guy never turns up at all, so perhaps that solves it.)

Hajek, Alan (2005), “The Cable Guy Paradox,” Analysis 65: 112-19.

Spell and Summon

This card curiosity is attributed to Lewis Carroll.

Lay down eight cards with these values:

Now add the values in each column, find a card of that value in the deck, and place it on top of the lower card. Aces count as 1, jacks as 11, queens as 12, and kings as 13. Thus in the first column 1 + 2 = 3, so you’d place a 3 on top of the 2.

When you’ve done all four columns, repeat the process, placing a 4 on the 3, etc. If a sum is more than 13, subtract 13 from it (for example, Q + 7 = 19 – 13 = 6). When you’ve exhausted the deck you’ll have four kings in the bottom row. Place each of these piles on the card above it, then take up the packs from right to left.

Turn the deck face down again and deal 13 cards in a circle, making note of which was dealt first. Counting from that card, deal 13 more cards, placing them on every second pile (that is, piles 2, 4, 6, etc.) and continuing around the circle until 13 are dealt. Then deal 13 more cards, one onto each third pile (piles 3, 6, 9, etc.), and finish by dealing the last 13 cards, one onto every fourth pile (4, 8, 12, etc.).

Each pile should now contain four cards. Take them up in order, starting with the first.

Now comes the payoff. Spell aloud A-C-E, dealing a card for each letter and turning the last one face up. It will be an ace. Continue with T-W-O, T-H-R-E-E, and so on up through J-A-C-K, Q-U-E-E-N, and K-I-N-G. In each case the last card will have the rank just spelled — and the full count will precisely exhaust the deck.

(From Robert Morrison Abraham, Winter Nights Entertainments, A Book of Pastimes for Everybody, 1932.)

The Infallible Seducer

Here’s a foolproof way to get anyone to sleep with you. Ask:

  1. Will you answer this question in the same way that you will answer the next?
  2. Will you sleep with me?

“If she keeps her word,” writes Richard Mark Sainsbury, “she must answer Yes to the second question, whatever she has answered to the first.”


zeno stadium paradox

Bertrand Russell explains Zeno’s paradox of the stadium:

Let us suppose three drill-sergeants, A, A′, and A′′, standing in a row, while the two files of soldiers march past them in opposite directions. At the first moment which we consider, the three men B, B′, B′′, in one row, and the three men C, C′, C′′ in the other row, are respectively opposite to A, A′, and A′′. At the very next moment, each row has moved on, and now B and C′′ are opposite A′. When, then, did B pass C′? It must have been somewhere between the two moments which we supposed consecutive. It follows that there must be other moments between any two given moments, and therefore that there must be an infinite number of moments in any given interval of time.

In other words, if time is a series of consecutive instants, and motion means passing through consecutive points, then the Bs are passing the As at the fastest possible speed — one point per instant. How then is it that the Bs are passing the Cs at twice this rate? It seems, Aristotle noted, that “half the time is equal to its double.”