Three Turning Rings
Read the puzzle
Three rings turn around a centre, each carrying one gem. At step 0 all three gems sit at the top marker. The outer ring has 9 evenly spaced notches, the middle ring 6 and the inner ring 4, and at every step each ring turns by one notch. At which step number are all three gems at the top again?
At which step number are all three gems at the top again?
36 STEPS
Notches 9, 6 and 4: the first common multiple is 36.
How to solve it
Step one: find when each gem returns on its own. The outer ring has 9 notches, so its gem is back at the top after 9 steps, then again at 18, 27, 36 and so on. The middle gem returns every 6 steps: 6, 12, 18, 24, 30, 36. The inner gem returns every 4 steps: 4, 8, 12, 16, 20, 24, 28, 32, 36.
Step two: find the first number that appears in all three lists. That is the least common multiple of 9, 6 and 4. Split them into primes: 9 is 3 times 3, 6 is 2 times 3, and 4 is 2 times 2. The smallest number that contains them all needs two 2s and two 3s: 2 times 2 times 3 times 3 equals 36.
Check it: 36 divided by 9 is 4 turns, 36 divided by 6 is 6 turns, and 36 divided by 4 is 9 turns, all whole numbers. So all three gems meet at the top at step 36.
The usual wrong answers are 18, which suits the outer and middle rings but leaves the inner gem halfway round, since 18 divided by 4 is 4.5, and 216 from multiplying 9 by 6 by 4, which works but is not the first time.
