Spatial and Sequence Reasoning: Patterns, Clocks, Cubes and Plans

Formulas (2)

• Arithmetic and geometric sequences · The nth term
  $$a_n = a_1 + (n - 1)\,d \qquad a_n = a_1\, r^{\,n - 1}$$
• Recurrence rules: each term built from the terms before it · Two-term recurrence
  $$a_n = p\,a_{n-1} + q\,a_{n-2}$$

Watch out for (3)

• The nth term is n minus 1 steps away
  → Arithmetic and geometric sequences
• Check the rule on every given term
  → Differences of differences, and interleaved sequences
• Use the term after the gap
  → Recurrence rules: each term built from the terms before it

Formulas (1)

• Letter codes: shifts and reversed alphabets · Shift code
  $$\text{coded position} = \text{position} + k \quad (\text{subtract } 26 \text{ if above } 26)$$

Reference tables (1)

Watch out for (2)

• Position from the end is 27 minus the position
  → Alphabet positions and letter sequences
• Shifts wrap from Z back to A
  → Letter codes: shifts and reversed alphabets

Formulas (3)

• Clock hands: rotations, degrees and elapsed time · Converting hand movement to time
  $$\text{minutes} = 60 \times (\text{minute-hand turns}) = 2 \times (\text{hour-hand degrees})$$
• The angle between the hour hand and the minute hand · Angle between the hands
  $$\theta = \lvert 30H - 5.5M \rvert \qquad (\text{if } \theta > 180°, \text{ use } 360° - \theta)$$
• Days of the week and times across midnight · Weekday after N days
  $$\text{shift} = N \bmod 7$$

Watch out for (3)

• Degrees of the hour hand are not minutes
  → Clock hands: rotations, degrees and elapsed time
• The hour hand moves between the hours
  → The angle between the hour hand and the minute hand
• Day 1 is the starting day
  → Days of the week and times across midnight

Formulas (1)

• Folding, punching and unfolding paper · Holes after folding and punching
  $$\text{holes} = \text{punches} \times 2^{f}$$

Reference tables (1)

Watch out for (2)

• A mirror reverses the order, not only the symbols
  → Letters and digits under mirrors and half turns
• A cut on the fold line makes one hole
  → Folding, punching and unfolding paper

Formulas (1)

• Painted cubes: how many small cubes have 0, 1, 2 or 3 painted faces · Painted n × n × n cube
  $$8 \;+\; 12(n-2) \;+\; 6(n-2)^2 \;+\; (n-2)^3 \;=\; n^3$$

Watch out for (3)

• "At least one face" includes the corners and edges
  → Painted cubes: how many small cubes have 0, 1, 2 or 3 painted faces
• Neighbours in the net are never opposite
  → Nets of a cube and finding opposite faces
• A view shows the tallest stack, not the total
  → Views of a stack of cubes: front, side and top

Formulas (3)

• Compass directions, turns and distance from the start · Distance from the start
  $$d = \sqrt{x^2 + y^2}$$
• Grids: counting shortest routes and reading numbered boards · Shortest grid routes
  $$\text{routes} = \binom{m + n}{n} = \frac{(m + n)!}{m!\, n!}$$
• Lengths along a line: fence posts and overlapping pieces · Overlapping pieces in a row
  $$\text{length} = n\,\ell - (n - 1)\,o$$

Watch out for (4)

• Right and left depend on the way you face
  → Compass directions, turns and distance from the start
• Count orders of moves, not moves
  → Grids: counting shortest routes and reading numbered boards
• Facing the centre flips left and right on the far side
  → Seating around a table: left and right when facing the centre
• Posts are one more than gaps on a line, equal on a loop
  → Lengths along a line: fence posts and overlapping pieces