Brain Teasers and Puzzles

Brain teasers, puzzles, riddles, and other challenges

Giant Cat Army Riddle from TED-Ed

In Dan Finkel’s TED-Ed video, he shares this math puzzle, paraphrased as follows:

Dr. Schrödinger is creating an army of giant cats for villainous purposes. Your team of secret agents has located his lab, but needs to get through his unusual security system.

The system displays a single number, and has three buttons that control this number:

  • Add 5
  • Add 7
  • Take the square root of the displayed number

Goal: make the numbers 2, 10, and 14 show on the display, in that order.

Rules:

  • The display starts at 0.
  • It’s fine if other numbers are displayed in between 2, 10, and 14, as long as they appear in that order.
  • The system will malfunction if any number is displayed more than once.
  • The system will malfunction if any number greater than 60 is displayed.
  • The system will malfunction if any fraction/decimal is displayed.

How can you achieve this?

(Sorry, the giant cat army has nothing to do with the puzzle.)

Source: Dan Finkel and TED-Ed

Continue reading “Giant Cat Army Riddle from TED-Ed”

Boy or Girl Paradox

The “boy or girl paradox” is a well-known brain teaser by famous puzzle-maker Martin Gardner. It’s popularly considered a “paradox” because (1) it has a highly unintuitive solution, and (2) its ambiguous wording meant either of two solutions could be valid solutions.

This is a rewording of that brain teaser to eliminate some ambiguity from that original question:

Out of all families with exactly two children, we randomly pick one family that has at least one boy. What is the probability that both children in this family are boys?

Assume only for the purposes of this puzzle that a child can only be a boy or a girl, and that either possibility is equally likely.

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Two Envelopes Paradox

The two envelopes paradox is a famous brain teaser of sorts, and not a true paradox. The problem is generally posed like this:

You are given a choice between two identical envelopes. One envelope contains some amount of money, and the other contains twice that amount of money. There is no way to distinguish between the two. However, when you choose one of the envelopes, before opening it, you are given the option of switching to the other envelope. Should you switch?

Why is this sometimes called a paradox? Well, if you choose to switch, you have a 50% chance of doubling your money, and a 50% chance of halving your money. If the amount of money in the envelope you initially chose is M, this reasoning suggests the expected amount in the other envelope is (2M + 0.5M) / 2 = 1.25M. This is more than M, so you should always switch.

But that would suggest once you’ve switched, you’re in the same position you were before you switched, so you should switch again. What is the problem with this reasoning?

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Friedman Numbers

Friedman numbers are positive integers that can be written as an expression of its own digits and basic arithmetic operations. While Friedman numbers don’t have much practical use, they can be fun and creative puzzles for all ages!

Rules of Friedman Numbers

  1. The expression must use the same number of digits as they occur in the Friedman number itself (e.g., an expression for 343 must use exactly two 3’s and one 4)
  2. Only these operations are allowed:
    • addition +
    • subtraction –
    • multiplication × or *
    • division ÷ or /
    • parentheses ()
    • exponents ^
    • concatenation (putting together multiple digits to make one number, such as using 1 and 2 to make 12)
  3. Trivial expressions are excluded (i.e., simply writing the number as itself)

Examples

Here are some of the smallest Friedman numbers and their expressions:

  • 25 = 52
  • 121 = 112
  • 125 = 5(1+2)

The Online Encyclopedia of Integer Sequences (OEIS) has a more comprehensive list of Friedman numbers and variations.

Challenges

Try to find expressions for these Friedman numbers:

1435 (easy)

Solution to 1435

35 × 41

2502 (easy)

Solution to 2502

2 + 502

28547 (hard)

Solution to 28547

(5 + 8)4 – (2 × 7)

123456789 (very hard)

Solution to 123456789

((2 × 7 + 86)5 – 91) / 34