$1.15 in Six Coins

A half dollar, a quarter, and four dimes.

You obviously can’t have any pennies, because you couldn’t have 1-4 pennies, since $1.15 is a multiple of 5, and if you had 5 pennies or more you could change a nickel.

You can’t have more than 1 nickel, otherwise you could change a dime. And if you had 1 nickel you couldn’t have but 1 dime, otherwise you could change a quarter. Then if you had 1 nickel and 1 dime you’d have 15 cents. That would leave $1.00 and you couldn’t make that remaining $1.00 with quarters and halves without either being able to change a half or a dollar.

So you can’t have any pennies or nickels.

Since you can’t have any pennies or nickels, you can’t possibly make $1.15 with dimes and halves only, so you must have at least 1 quarter. You can’t have more that one quarter, otherwise you could change a half. So you must have exactly 1 quarter.

That leaves 90 cents, to make with a half dollar and dimes. So that’s 1 half and 4 dimes. And with only 4 dimes you can’t change a half dollar.