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Quant Finance · Topic lesson

Brainteasers

The two mechanisms behind most puzzle questions, with live numbers: price an option to try again by working backwards, and turn a rate puzzle into one division by holding the relative speed. The classic puzzles that do not change are on the Concepts page.

2 chapters About 21 minutes0 of 2 complete
Start chapter 1

Brainteasers are not tests of whether you have seen the puzzle. They are tests of whether you reach for a mechanism or try to remember an answer. Most of them reduce to one of a few moves, and two of those moves come with numbers you can change: valuing an option to try again, which is backward induction and prices American options; and rate puzzles, which are relative speed and answer clock questions, catching-up questions and 'how many times' questions.

The method from the old lesson survives here as the way to attack anything unfamiliar. Restate the rules and test the edge cases before calculating. Simplify to a small case you can count by hand. Find the mechanism. Then verify against a limiting case: no rerolls, the top of the hour, a coin that always lands heads. Fifty-five classic puzzles that do not parameterize, from the socks in the drawer to the two eggs, live on the Concepts page under Brainteasers with model answers.

  • Dice reroll. The average face, the keep rule, the value with one, two and three rerolls, what each reroll adds, the die's size from the value.
  • Clock angles. Where the hands are, the angle folded into 0°–180°, the next overlap and right angle, why 11 meetings per 12 hours, the angle ten minutes later.
The rule that solves most puzzles

Find the quantity that stays fixed while everything else moves, and work from it. In the reroll game it is the continuation value: what a fresh roll is worth. On the clock it is 5.5° a minute. Once you hold that number, the puzzle is one comparison or one division.

How to attack a puzzle you have not seen

Each step points to the chapter that practices it.

  1. 1
    Restate the rules

    Final roll binding? Smaller angle? Strictly after?

    Chapter 1Chapter 2
  2. 2
    Find the fixed quantity

    The continuation value; 5.5° a minute.

    Chapter 1Chapter 2
  3. 3
    Work backwards

    From the last decision to the first.

    Chapter 1
  4. 4
    Divide the gap by the rate

    Overlap, right angle, opposite.

    Chapter 2
  5. 5
    Verify at a limit

    No rerolls; the top of the hour; 12:00.

    Chapter 1Chapter 2

Chapters

1

What is a second try worth? Price it backwards

11 min

A die pays its face; you may reroll. The option to try again is worth money, and the way to price it is from the last decision backwards: keep what beats the continuation, throw back what does not.

  • Value a die with no reroll, then with one, by the keep-or-reroll rule
  • Chain the rule for two and three rerolls
  • Explain why each extra reroll is worth less than the last
  • Recover the size of the die from the value of the one-reroll game
2

Clock hands: relative speed in a costume

10 min

The minute hand gains 5.5° a minute on the hour hand. Hold that number and every clock question is one subtraction or one division.

  • Find the angle between the hands at any time, remembering the hour hand moves
  • Find the next time the hands overlap, or make a right angle, by dividing a gap by 5.5
  • Explain why the hands meet 11 times in 12 hours, not 12
  • Handle the fold: angles are reported between 0° and 180°