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 1 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 64. The easiest way to find the solution is to use trial and error, looking at the ascending cubes to see whether they’re also perfect squares. We can rule out 2³, because the square root of eight is not an integer. Likewise, 3³ doesn’t work. But when we try 4³, we get 64, which has a square root of 8.

There’s another way to think about the problem, though. A different way to represent some numbers that are both perfect squares and perfect cubes is as a number with a base that is both squared and cubed. When you raise a power to a power, you multiply the exponents, so we can write this as a base raised to the sixth power. Let’s look at our original example: (1²)³ = (1³)² = 1^{6} = 1. So to find the next smallest number that is both a perfect square and a perfect cube, we can simply try the next smallest base: (2²)³ = (2³)² = 2^{6} = 64. This leads to a natural question. Can every number that is both a perfect square and a perfect cube be written as an integer to the sixth power?

**2.** The bee will have traveled 64 miles. There’s a little trick involved here. We don’t need to add up all the distances that the bee flies back and forth between the cars. We just need to know the amount of time that the bee has been flying. If the two cars start out 60 miles apart and move at the same speed, they’ll each travel 30 miles. Since they’re both going 15 mph, they’ll both take two hours to go that far. A bee flying 32 mph for two hours travels 64 miles.

This problem is a variation of a problem once posed to the great mathematician John Von Neumann, who was famous for his ability to do fantastically hard problems in his head. When someone posed the problem to him, he immediately gave the correct answer. “Oh, you must have heard the trick before!” the person is said to have exclaimed. “What trick?” Von Neumann responded. “All I did was sum the infinite series.”

**3.** This puzzle has more than one solution (there are 12 not counting rotation, in fact), but they all share the concept of pebbles being an L-shaped distance from one another. Here’s one of the the solutions:

I find this solution to be the easiest and most natural of the 12. Is it the one you found? If you feel up to it, I encourage you to try to find all 12!

This problem has a rich background. It is known as the Eight Queens Problem. It has served as a test problem in computer science because of the large number of ways you can place eight queens on a chess board. How does a computer find a solution to a problem with so many possible configurations, but with no set formula?

Taking this even further, some have tried to see if you can solve the same problem with *n* pebbles on an *n-*by-*n* grid. Of course, you can do this simply with *n* = 1. What’s the next number for which a solution exists?