Last week, in Fundamental Understanding of Mathematics X, we took a look at the Distributive Property of Addition over Multiplication, a very fancy way of saying that when you multiply two numbers, it really doesn't matter if one of the numbers is an actual value or a sum of two values.
This week, though, I am rushed and will not be around Saturday morning to comment. The rush involves replacing my old, broken down computer with a brand spanking new screamer of a beast which just came back from the shop on Thursday all shiny and spinny and empty. The not being around Saturday involves some work related training.
So, consider this to be something along the lines of, "Class, I'll be out for a bit, talk quietly among yourselves."
But, I think you'll all appreciate something mathematical to talk about. Since we had a very interesting discussion in the comments about the notorious three door Monty, I thought today I'd to throw a few more interesting problems at the wall, and see if any of them stick.
First Question: XX + YY + ZZ = YXZ. What are X, Y and Z? (note: treat X, Y, Z as if they are digits: if X is one, XX is eleven)
Second Question: A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?
Third Question: Some relationships are transitive, if A>B and B>C then A>C, but some relationships are intransitive , for example if Ann likes Betty and Betty likes Carole it does not necessarily follow that Ann likes Carole.
When voting, if Abel beats Baker and Baker beats Charlie we might reasonably expect Abel to beat Charlie. But consider a voting system where candidates are ranked in order of preference. Suppose we have three candidates and three voters who cast the following votes: ABC, BCA and CAB.
In this case,
A beats B, 2 to 1
B beats C, 2 to 1
C beats A, 2 to 1
We call this an intransitive outcome. If three voters vote in this election, and each voter ranks all three candidates, how many outcomes are possible, and how many are intransitive?
[edit] An interesting extension to this: given that intransitive election outcomes are a bad thing, what structural change to this system would eliminate intransitive outcomes?[/edit]
Fourth Question: Three apples and two bananas cost $1.29. Two apples and three bananas cost $1.76. You want one apple and one banana. What's the price?
Have fun in the comments.