The Dominoes and the Chessboard

On Monday, I posted a puzzle regarding a chessboard and some dominoes. As promised, I am now going to give the answer (so that my boyfriend's family will stop arguing about it!)

So, is it possible to place 32 dominoes, each covering two squares, on a chessboard with two diagonally opposite corners cut out?

The answer is: No. It is not possible.

But how do you prove it? Most of the modules in my degree this term are all about proving things, and it can get very confusing. Especially Analysis. But there is a simple way to think about this puzzle.

As the two corners cut out of the chessboard are diagonally opposite each other, they must be the same colour. This means that we have 30 of one colour (in the picture above it is black) and 32 of the other colour (obviously, in this case white). As the dominoes only cover two squares, no matter how you place them, they are going to cover one of each colour. So if we pair off the colours, we can match 30 black to 30 white, but then we are left with 2 of the same colour left over. We know that a single domino cannot cover two squares of the same colour, so the puzzle is therefore impossible.

Hopefully I explained that clearly so you aren't left feeling as confused as before!

cos^2(θ) + sin^2(θ) = 1

I was doing an A-level maths paper when I found this question:

Use this triangle to prove cos²θ + sin²θ = 1. For what values of θ is this prove valid?

 My feelings about proof questions are conflicted. When I can see how it works, and I can find an elegant solution, I absolutely love them. Being able to show why something works cements the idea to me that there is a reason for why formulas work for any value you put in, or why you can substitute one formula for another and still come out with the same answer. However, when I can't see how all of the pieces fit together to make a proof, it really frustrates me. I guess that's part of being a mathematician, not knowing how the pieces fit together to start with, but persevering with it until they do.

As it happens, this is a proof question that I could fit together, so I am going to share it here. In my opinion, this is quite an easy one, but easy is subjective, so maybe someone will find it impressive anyway XD

1. Because this triangle is right-angled, I can use trigonometry to show that cosθ = b/c and sinθ = a/c

2. If I square both of them, I end up with cos²θ = b²/c² and sin²θ = a²/c²

3. Next I add the two together to give cos²θ + sin²θ = (a² + b²)/c²

4. Pythagoras says: a² + b² = c² so (a² + b²)/c² = 1

5. Therefore cos²θ + sin²θ = 1

For the second part of the question, the proof is only valid for θ < 90^o because if θ > 90^o, ABC wouldn't be a right-angled triangle.

So that's all there is to the proof. Like I said, I think that one is quite nice. And now to leave you with an interesting fact about the Pythagoras Theorem. In China, it's named after mathematician Shang Gao, who came up with it independently from Pythagoras about 2,200 years ago (300 years after Pythagoras).