Math Made Easy: Problem of the Day 24
One technique for easily solving math problems I’ve used so far, but haven’t really talked about in the Problems of the Day, is drawing. Enough can’t be said about how helpful making a picture of the situation is. Many times, just making the drawing will make you see the solution to the problem. If you can avoid unnecessary computation in your math, you can save time, and for many of us, a headache. Let’s see how we can use this to solve today’s problem.
You could just try dividing by two repeatedly until you get to 1, and try not to lose track of what you’re doing. Or you could try to remember a formula for figuring this out. Or, you can draw a picture. A great way to handle this is a tree diagram. The top of the tree is the 209 teams. Split it in two every time. On one branch, terminate with the number of teams who lost. The other branch will continue on, and it will have the number of teams who won, plus any team that may have had a bye. There should only be one bye team in any given round, and the continuing branch should always have the higher number. If I use this technique, I produce a tree like this:
It took 8 rounds to find a champion. Note the 209 at the very top of the tree doesn’t count as a round. Only each split does.
Another method to do this is to count powers of two until you find the first one bigger than 209. It turns out that is 2^{8}, which is 256. The exponent tells me how many rounds it took, which is 8 rounds, just like the tree showed me.
The benefit of drawing a picture, is that, when you don’t KNOW how to do a problem – don’t know the mathematics behind it, or whatever formulae you might need, the geometry of the situation can frequently solve the problem for you in the process of drawing it, just like it did here. (It doesn’t look like geometry, but it is once you find a way to graphically represent it).
