Spatial Reasoning, Clocks and Calendars

Painted cubes

A large cube painted on all faces is cut into n × n × n small cubes. Count by position:

For n = 4: 8 + 24 + 24 + 8 = 64 cubes. For n = 5: 8, 36, 54 and 27.

Counting figures in a grid

In an n × n grid of unit squares:

• Squares of all sizes: 1² + 2² + … + n² = n(n + 1)(2n + 1)/6. A 4 × 4 grid has 30; a chessboard (8 × 8) has 204.
• Rectangles (including squares): choose 2 of the n + 1 vertical lines and 2 of the n + 1 horizontal lines: C(n + 1, 2)². A 4 × 4 grid has 100; a chessboard has 1296.

Mirror images, rotations and paper folding

• A mirror reflects left and right (for a vertical mirror) but not top and bottom. In a mirror, a clock showing 3:00 looks like 9:00; the mirror time is 12:00 − actual time (11:60 − actual time for minutes).
• A rotation keeps the figure's handedness; check where one distinctive mark goes and whether the order of the others (clockwise or anticlockwise) is preserved.
• Paper folding and punching: each fold doubles the number of holes; unfold step by step, reflecting the holes across each fold line.

Clocks

The minute hand moves 6° per minute and the hour hand 0.5° per minute (30° per hour). At h hours and m minutes, the angle between them is

|30h − 5.5m| (subtract from 360° if it exceeds 180°).

• At 3:40: |90 − 220| = 130°.
• At 9:15: |270 − 82.5| = 187.5°, so the smaller angle is 172.5°.
• The hands overlap every 720/11 ≈ 65.45 minutes, which is 11 times in 12 hours (not 12).

Calendars

• An ordinary year is 52 weeks + 1 day, a leap year 52 weeks + 2 days: the same date moves 1 weekday forward each year, 2 after a 29 February.
• Leap years: divisible by 4, except century years, which must be divisible by 400 (2000 was a leap year, 1900 was not).
• The odd days method counts the remainder of the number of days modulo 7.

Python's datetime gives quick checks:

import datetime
from math import comb

def painted_cubes(n):
    return {"3 faces": 8, "2 faces": 12 * (n - 2), "1 face": 6 * (n - 2) ** 2, "0 faces": (n - 2) ** 3}

def clock_angle(h, m):
    a = abs(30 * h - 5.5 * m) % 360
    return min(a, 360 - a)

print("4x4x4 painted cube:", painted_cubes(4))
print("squares in 8x8:", sum(k * k for k in range(1, 9)), "| rectangles:", comb(9, 2) ** 2)
print("angle at 3:40 =", clock_angle(3, 40), "| at 9:15 =", clock_angle(9, 15))
print("15 August 2047 is a", datetime.date(2047, 8, 15).strftime("%A"))

4x4x4 painted cube: {'3 faces': 8, '2 faces': 24, '1 face': 24, '0 faces': 8}
squares in 8x8: 204 | rectangles: 1296
angle at 3:40 = 130.0 | at 9:15 = 172.5
15 August 2047 is a Thursday

Practice questions

import datetime
print(datetime.date(2031, 1, 1).strftime("%A"))