TwiddleLab / Courses / Math
Counting and Patterns
Count choices with trees, spot the rule in sequences and topple proofs like dominoes.
Best after: Algebra in Motion
Lessons
- Choices Multiply (5 min): When choices are made one after another and do not limit each other, multiply the number of options.
- Arrangements: Factorials as Trees (7 min): An arrangement (permutation) is an order. n different things can be ordered in n! ways.
- Choosing a Team (7 min): When order does not matter, it is a combination: a team, a pair, a handful.
- Pascal's Triangle (7 min): Routes that go only right or up: the count at a crossing is the sum of the counts from the left and from below.
- Sequences: Spot the Rule (7 min): An arithmetic sequence adds the same amount each step: 4, 7, 10, 13... Term n is first + (n - 1) × jump.
- Adding Up a Sequence (7 min): Two copies of the staircase 1 + 2 + ... + n make an n by (n + 1) rectangle.
- Infinite Sums That Stop (7 min): 1/2 + 1/4 + 1/8 + ... never ends, but the total gets as close to 1 as you like and never passes it.
- Proof by Dominoes (7 min): Checking many examples is not a proof: 1, 2, 4, 8, 16 regions, and then 31.
- Pigeonholes (7 min): Pigeonhole principle: put more than n things into n boxes, and some box gets at least two.
- Clock Arithmetic: Remainders (7 min): On a dial with n spots, only the remainder after dividing by n matters: 100 hours on a 12-hour clock moves the hand 4.
- Project: How Many Passwords? (8 min): A password rule is a chain of choices: choices per spot, multiplied once per spot. 8 spots from 62 characters is 62 to the power 8.
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