TwiddleLab / Math / Counting and Patterns
Clock Arithmetic: Remainders
What you will learn
- 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.
- We write 100 mod 12 = 4, and say 15 and 3 are the same mod 12. In Python: 100 % 12.
- Weekdays work mod 7, a Caesar cipher works mod 26.
- Jumping by 5 round 12 visits every spot; jumping by 4 gets stuck on 0, 4, 8.
How the lesson goes
It's 9 pm. Your game event starts in 100 hours.
What time will it be? Drag the hour hand to your bet.
Wind the clock: exactly 15 hours, then exactly 30.
Drag the hand round clockwise. The counter keeps the total.
Now the hook: 100 hours from 9 pm.
Wind whole laps while you can, then move the rest.
This is modular arithmetic.
Divide and keep only the remainder: 100 = 8 × 12 + 4.
We write 100 mod 12 = 4, and 15 ≡ 3 (mod 12): same spot on the dial. In Python it is 100 % 12.
Adding and multiplying still work: just wrap round at the end.
Put each number on its spot on a 0 to 11 dial.
Drag the balls. Laps don't count.
Jump round the dial: 5 spots at a time, from 0.
Tap where each jump lands. Then the dial finishes the pattern.
Game: what day will it be?
Today is Monday. Tap the day for each jump. Weeks wrap round mod 7.
A Caesar cipher shifts every letter round the alphabet.
Turn the inner ring to your own shift, pick a word, then lock your key.
What does this Python line print?
Four 12s make 48, and 50 - 48 = 2, so 50 % 12 is 2.
Remainders wrap numbers round a dial.
A message arrived, locked with your key. Turn the ring back to read it.
Previous lesson: Pigeonholes · Next lesson: Project: How Many Passwords? · Part of the course: Counting and Patterns
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