The Mechanic:

Every player has a strip of cardboard on which there are slots numbered left to right from 6 to 0. Players take turns drafting tiles from a general pool. Each tile has a number on it. When you draft a tile, you place it under the slot corresponding to the number on said tile. Then, you slide all of your tiles down one slot. Whenever a tile hits 0, its action is triggered and then it is placed back under the slot corresponding to its number. For example, a number 1 tile will trigger every turn, whereas a number 4 tile will trigger every four turns. Of course, the higher numbered tiles will have more powerful effects.

The Paradox:

I was playing around with this idea a little bit last night, and started recording how many turns it would take for two differently-numbered tiles placed in the same slot to arrive back in the same slot. For example, it takes 15 turns for a 5 and 3 tile originally placed in the same slot to re-align.

The pattern seemed to be very simple at first: You simply multiply the two numbers on the tiles, and this gives you the number of turns.

However, there are exactly four exceptions:

- 2 and 4 will realign in 4 turns

- 2 and 6 will realign in 6 turns

- 3 and 6 will realign in 6 turns

- 4 and 6 will realign in 12 turns

The Question:

Can anyone make sense of why this is the case?

Hahaha, lol I can't believe I didn't figure this out... evidently math isn't my strong suit!

Yeah, the case with the number 5 is certainly interesting... I don't think it will be an issue though if I design number 5 tiles with this in mind...

Thanks for your fresh pair of eyes!