# Math on the Beach

Sunday at the beach, Simon was reenacting the 5 doors and a cat puzzle (he had learned this puzzle from the Mind Your Decisions channel). The puzzle is about guessing behind which door the cat is hiding in as few guesses as possible, while the cat is allowed to move one door further after every wrong guess.

“Here’s a fun fact!” Simon said all of a sudden. “If you add up all the grains of sand on all the beaches all over the world, you are going to get several quintillion sand grains or several times 10^18!” He then proceeded to try to calculate how many sand grains there might be at the beach around us…

In the evening, while having a meal by the sea, Simon challenged Dad with a Brilliant.org problem he particularly liked:

Simon’s explanation sheet (The general formulas are written by Simon, the numbers underneath the table are his Dad’s, who just couldn’t believe Simon’s counterintuitive solution at first and wanted check the concrete sums. He later accepted his defeat):

# Symmetric Chaos

“What you’ll find is that the path is chaotic while the points are actually symmetric,” – Simon tells me, graphing a sequence on the beach. (On the picture below, the sequence graphed is the one on the right):

# Math on the Beach

Simon doing math everywhere.

And he showed me this beautiful trick of two rows adding up to equal numbers and their squares adding up to equal numbers. And the two rows below? Even their cubes!

Now, can you come up with two rows in which also the fourth powers add up to equal sums?

Simon learned this trick from Matt Parker: you should pick numbers up to n-1, where n is the next power of 2. In this case, n would be 2 to the fifth power and that is 32, so we pick numbers up to 31. Then we write them down in two rows in such a way that the top row only has numbers whose binary expressions have an even number of ones and the bottom row – only odd number of ones.

Simon also came up with an interesting fact about the trick using a pattern of “buckets” turned in opposite directions:

# High and low tide

We drew a line to mark how far the sea has pulled back between 17:30 and 21:00 o’clock. It’s amazing to observe the the Moon in action! We have read about this dance that the Earth and the Moon dance with each other, circling round and round in what is currently our favorite book, Eine kleine Nachtphysik by Wolfgang Rosler: