No Tabs

William West noticed this inscription in an alehouse near Brighton. What does it mean?

Click for Answer

Digit Sums

Sort the numbers 0, 1, 2, …, 123456 into two sets. In one set put all the numbers who digits add to an even sum; in the other put those whose digits produce an odd sum. Which set is larger?

Click for Answer

Work Planning

A logic exercise by Lewis Carroll — what conclusion can be drawn from these premises?

  1. I despise anything that cannot be used as a bridge.
  2. Nothing that is worth writing an ode to would be an unwelcome gift to me.
  3. A rainbow will not bear the weight of a wheelbarrow.
  4. Whatever can be used as a bridge will bear the weight of a wheelbarrow.
  5. I would not take, as a gift, a thing that I despise.
Click for Answer

Times Square

Prove that the product of four consecutive positive integers cannot be a perfect square.

Click for Answer

Word Ladders

On Christmas Day 1877, assailed by two young ladies with “nothing to do,” Lewis Carroll invented a new “form of verbal torture”: Presented with two words of the same length, the solver must convert one to the other by changing a single letter at a time, with each step producing a valid English word. For example, HEAD can be converted to TAIL in five steps:


Carroll called the new pastime Doublets and published it in Vanity Fair, which hailed it as “so entirely novel and withal so interesting, that … the Doublets may be expected to become an occupation to the full as amusing as the guessing of the Double Acrostics has already proved.”

In some puzzles the number of steps is specified. In Nabokov’s Pale Fire, the narrator describes a friend who was addicted to “word golf.” “He would interrupt the flow of a prismatic conversation to indulge in this particular pastime, and naturally it would have been boorish of me to refuse playing with him. Some of my records are: HATE-LOVE in three, LASS-MALE in four, and LIVE-DEAD in five (with LEND in the middle).” I’ve been able to solve the first two of these fairly easily, but not the last.

But even without such a constraint, some transformations require a surprising number of steps. Carroll found that 10 were required to turn BLUE into PINK, and in 1968 wordplay expert Dmitri Borgmann declared himself unable to convert ABOVE into BELOW at all.

In a computer study of 5,757 five-letter English words, Donald Knuth found that most could be connected to one another, but 671 could not. One of these, fittingly, was ALOOF. In the wider English language, what proportion of words are “aloof,” words that cannot be connected to any of their fellows? Is ALOOF itself one of these?

In 1917 Sam Loyd and Thomas Edison made this short, which plays with similar ideas. The goat at the end was animated by Willis O’Brien, who would bring King Kong to life 16 years later:

Ends and Means

Prove that if each point in the plane is colored red, yellow, or blue, a unit segment must exist whose endpoints are the same color.

Click for Answer