# A Coin Puzzle: My Two Cents (Plus 97 More)

Our friends at Penny Dell Puzzles recently shared the following brain teaser on their social media:

Naturally, we accepted the challenge.

Now, before we get started with this one, we have to add one detail: which coins we’re allowed to use. It’s safe to assume that pennies, nickels, dimes, and quarters are available, but the question doesn’t say anything about half-dollar coins.

So we’re going to figure out the correct answer without half-dollar coins available, and then with half-dollar coins available.

Let’s begin.

[Image courtesy of How Stuff Works.]

The easiest way to get started is to figure out the smallest number of coins we need to make 99 cents, since that’s the highest number we need to be able to form. Once we have that info, we can work backwards and make sure all the other numbers are covered.

For 99 cents, you need 3 quarters, 2 dimes, and 4 pennies. That’s 25 + 25 + 25 + 10 + 10 + 1 + 1 + 1 + 1 = 99.

Right away, we know we’re close with these 9 coins.

You don’t need more than 3 quarters, for instance, because your possible totals are all below \$1.

Now, let’s make sure we can form the numbers 1 through 24 with our chosen coins. (If we can, we’re done, because once we’ve covered 1 through 24, we can simply add one quarter or two quarters to cover 25 through 99.)

Our four pennies cover us for 1 through 4. But wait, there’s 5. And we can’t make 5 cents change with 4 pennies or 2 dimes. In fact, we can’t make 5, 6, 7, 8, or 9 cents change without a nickel.

So let’s add a nickel to our current coin count. That makes 3 quarters, 2 dimes, 1 nickel, and 4 pennies. (Why just 1 nickel? Well, we don’t need two, because that’s covered by a single dime.)

Our four pennies cover 1 through 4. Our nickel and four pennies cover 5 through 9. Our dime, nickel, and four pennies cover 1 through 19. And our two dimes, one nickel, and four pennies cover 1 through 29. (But, again, we only need them to cover 1 through 24, because at that point, our quarters become useful.)

That’s all 99 possibilities — 1 through 99 — covered by just ten coins.

[Image courtesy of Wikipedia.]

Well, we can apply the same thinking to a coin count with a half-dollar. For 99 cents, you need 1 half-dollar, 1 quarter, 2 dimes, and 4 pennies. That’s 50 + 25 + 10 + 10 + 1 + 1 + 1 + 1 = 99.

Now, we make sure we can form the numbers 1 through 49 with our chosen coins. (Once we can, we can simply add the half-dollar to cover 50 through 99.)

Once again, we quickly discover we need that single nickel to fill in the gaps.

Our four pennies cover 1 through 4. Our nickel and four pennies cover 5 through 9. Our dime, nickel, and four pennies cover 1 through 19. Our two dimes, one nickel, and four pennies cover 1 through 29. And our one quarter, two dimes, one nickel, and four pennies cover 1 through 54. (But, again, we only need them to cover 1 through 49, because at that point, our half-dollar becomes useful.)

That’s all 99 possibilities — 1 through 99 — covered by just nine coins.

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