Here's a different perspective on the question :

Each time Jamie gives 1 bead to Amy, Jamie's total decreases by 1. It might come to our minds at first that their difference changes by 1 only.

But Amy's total increases by 1 as well. This means that their difference actually changes by 2 each time Jamie gives Amy 1 bead.


So, from 288 more beads to 336 fewer beads than Amy,

288 + 336 = 624
The difference changed by 624.


624 ÷ change of 2 per bead given = 312

Jamie gave 312 beads to Amy.

312 is therefore 2/3 of Jamie's beads, which is like saying 2 units out of 3 units.

1 unit = 312 ÷ 2 = 156
3 units = 156 x 3 = 468

(Or in a single step, 312 x 3/2 = 468)

So Jamie has 468 beads at first.

Amy has 468 - 288 = 180 beads at first.