Solutions:

1. If the three checks are made out in the amounts of $15, $30, $60, then you reduce the maximum change left with Claude to $14 no matter what the price.

2. Claude might publish $50, $51, $53, $59. Suppose you could find a set of checks that eliminates all change no matter which item your teenager selects. Some check must be $50 or less. If $50, then another must be $1, so the third must be $8, because every item must be purchasable. In that case the $53 would require change. If Claude has no check for $50, but one for a lesser amount x, then Claude requires a check for $50 - $x. The remaining check must reach $59, so $51 and $53 will require change.