To use the least number of boxes, we will need to pack as many puffs as possible into boxes of 10 and minimum number of puffs in boxes of 6.

Since 408 is not perfectly divisible by 10, we will need to find the next largest number that is perfectly divisible by 10 with the difference being multiples of 6 (being able to pack into boxes of 6).

1 boxes of 6: 1 x 6 = 6
Left: 408 - 6 = 402 (not perfectly divisible by 10)

2 boxes of 6: 2 x 6 = 12
Left: 408 - 12 = 396 (not perfectly divisible by 10)

3 boxes of 6: 3 x 6 = 18
Left: 408 - 18 = 390 (perfectly divisible by 10)

Least number of boxes
= 39 boxes of 10 puffs + 3 boxes of 6 puffs
= 42 boxes