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A Square and A Cube

The big picture

Here are all ten ideas, in order.

A Square and A Cube — the whole chapter on one screen
IdeaRememberExample
Square numberSame count across as down.7² = 49
Odd staircaseEach new layer is the next odd number.16 + 9 = 25
Ruling outSquares end in 0, 1, 4, 5, 6 or 9 only.72 is not a square
Square rootThe side the count came from. Check by multiplying back.√64 = 8
Pair the primesBreak, pair, keep one per pair, multiply.36 → 2 × 3 = 6
In betweenFind the square below and the square above.20 → between 4 and 5
Cube numberA stack as deep and tall as it is wide.3³ = 27
Cube patternsCubes end in every digit. Odd groups add up to cubes.7 + 9 + 11 = 3³
Group in threesThe pairing method, in groups of three.2744 → 14
Which countCovering a square patch: side × side. Filling a cube-shaped space: side × side × side.4 units → 16 or 64

Chapter 1 of Part I is A Square and A Cube. Read it next and find these ideas in it.

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This track

  1. 0 Start here
  2. 1 A number you can draw as a square
  3. 2 Each new layer is the next odd number
  4. 3 A test that can only say no
  5. 4 Going back to the side you started from
  6. 5 Pairing the primes finds the side
  7. 6 Checkpoint: squares, the staircase and roots
  8. 7 When no whole number fits
  9. 8 A cube number counts real cubes
  10. 9 What cubes end in, and what they add up to
  11. 10 Grouping the primes in threes
  12. 11 Which count is the job asking for
  13. 12 Checkpoint: in between, cubes and cube roots
  14. 13 The big picture
  15. 14 Track quiz

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