Below, you’ll find the answers to last week’s puzzles. For those who didn’t get a chance, if you would like to take a crack at the Week 2 challenge, just click here.

I hope you noticed what all of these questions had in common. I’ll try to include a different theme each week.

**1.** The answer is 44 cents. Think of it this way: since the question requires us to use at least seven coins, we can assume we’re going to use only seven coins. And since we’re looking for the smallest amount of money, we’ll want to maximize the number of pennies. You might think, then, that the answer is seven cents (seven pennies). But of course you could exchange five of those pennies for a nickel and so represent seven cents with three coins. What’s the largest number of pennies that you *can’t* substitute another coin for? Four pennies. So we have four pennies and three coins to go. The next smallest coin, of course, is a nickel. What if we try to use two nickels? That’s a dime. We can only use one nickel before we have to reach for a dime. So now we have four pennies and a nickel, with two coins to go. Add a dime for the sixth coin. What if we added two dimes? Well, two dimes plus a nickel is the same as a quarter, which bumps us back to five coins. So we’ll leave the dime at one. One coin to go, and it can’t be a penny, nickel, or dime. What’s left? A quarter. So we have four pennies, a nickel, a dime, and a quarter — seven coins adding up to 44 cents.

**2.** It would take a quarter of a minute to guarantee there are no ants left on the stick. One easy approach is to imagine dots instead of ants moving toward each other. If you saw two identical dots moving toward each other at the exact same speed, and they lost no time when they collided, it would be impossible to tell whether the two dots had reversed direction or whether they had just passed each other and were continuing forward along the line. Since we’re only concern about clearing the stick, we can think of the ants as passing each other instead of colliding and reversing direction. How long does it take for the ants to traverse a one-foot stick traveling at four feet per minute? One quarter of a minute, or fifteen seconds.

**3.** The probability that the other coin is also heads is 1/3. Some people probably assumed that the answer is 1/2, or fifty percent. After all, a flipped coin has a 50 percent chance of landing heads, and flipping one coin doesn’t affect the outcome of the second coin. Each coin flip is independent. But we need to look at the chance of two probabilistic events happening together. Let’s call one coin A and the other coin B, and let’s assume, for a moment, that we don’t know that one of the coins has landed heads. What are the possible outcomes? That’s easy enough to determine:

A: Heads B: Tails

A: Tails B: Heads

A: Heads B: Heads

A: Tails B: Tails

Now we can return to the information we have: one of the coins is heads, and we want to know the chance that the other coin is also heads. We can rule out the last, since already we know that one of the coins has landed heads. One out of the three remaining options is two heads, and so the answer is 1/3.