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:
It’s been great watching Simon turn a word problem into an equation today. He was busy with “Sums of consecutive integers” (Practice finding the nth number in a sequence of consecutive numbers based on the sum) on Khan Academy.
Signed Simon up for Brilliant, a service that e-mails you quick math problems every day. It took Simon about 5 minutes to solve this sequence. Longer than myself, I must admit. First he jotted some guesses on a sheet of paper…
But then he yelled: It’s to the power of! 2 to the power of 0 is one, to the power of 1 is 2 – that’s how we get 5, and so on. 2 to the power of 5 is 32. 33 + 32 = 65, the correct answer!
The same evening while having dinner he all of a sudden thought of a second way to solve the same sequence. He didn’t even want to eat his dinner any more as he said he enjoyed continuing the sequence so much. He was very quick at adding hundreds and thousands.
He said he could not wait until we got a new task the next morning!
This is a perfect example of how rows of boring sums are totally not motivating at school and the child is slow and unwilling to solve them, but when doing sums is part of something very exciting – solving a personal difficult and exciting math challenge that you’re being e-mailed every day it’s a whole different story.