At Arnaldo’s small party, people shook hands with as many people they liked, from 0 up to everyone else at the party. Nayla has been paying close attention, and says that if you add up everyone’s personal “handshake total”, you will get 15. Can she be correct? Explain.
Often in life, when someone says something that seems surprising, counter-intuitive, or even reasonable, we will look not at the conviction with which the speaker is speaking, but instead for the evidence and reasoning needed to back up their claim. In mathematics, we do the same thing. It is a fun fact of mathematics that many things that initially appear to be false are in fact true, and as well many things that appear to be true are false. So the only way to know if a statement is correct is to use clear reasoning based on assumptions we all agree upon.
Proving is a way of demonstrating to yourself and others that something you think may be true (or false), must in fact be true (or false). Although there are many different ways of proving a statement, everyone agrees that the steps of a proof should be clear: a careful reader should be able to read each step of your proof and see how it connects and follows from the one before. That way, at the conclusion of your proof, they will agree with you!
In the “Arnaldo’s party” problem above, at first it may seem a bit mystifying how to approach testing Nayla’s claim. Probably the best thing to do at first is to Tinker by trying to see if you can get a “handshake total” for all party members to be equal or close to 15. If you can get to exactly 15, you’re done; Nayla is correct, and the problem is over. On the other hand, if you have tried a number of different handshake “scenarios” and only gotten close to 15 in each case, nevertheless you will probably have noticed something interesting about handshake totals in general—something you might try to prove (and which will tell us if Nayla can be correct as well).
So, when presented with a claim, do three things. First, decide whether it seems reasonable – do you believe it at first glance? Next, try to find evidence that the claim is or is not true. Finally, for any claim that you believe to be true, try to prove that it’s true by making a sequence of logical statements, each of which your peers could understand and agree to; and for any claim you believe to be false, provide an example that shows the claim is false—such an example is called a counterexample.
Here’s a problem that many students have enjoyed:
A 64-square black and white checkerboard (see above) can clearly be completely covered by 32 dominos. If the top right and bottom left squares are removed, what is left is a 62-square board. How many dominoes will be required to cover the 62-square board?
[Continuation of problem 2] Did you check your answer to problem 2? A 64 square checkerboard is covered by 32 dominoes since every domino covers 2 squares, a white one and a black one. In the altered checkerboard, how many black squares are there? How many white squares? Would 31 dominoes do the job?
Here’s another problem that might seem very tricky at first, but with a little persistence and some tinkering . . .
In a stock market game, Traci claims that she can purchase any stock above $\$26$ by using only $\$5$ and $\$6$ coupons. (For example, there is no way she could purchase $\$13$ of stock with these coupons, but she could purchase $\$41$ of stock by using one $\$6$ coupon and seven $\$5$ coupons.) Do you think that Traci can really pull this off?
As a token of our affection to you, the answer is given on the reverse.
While it may seem plausible that there is always some combination of $\$5$ and $\$6$ coupons that is equivalent to a specific amount of money, it at first seems unlikely that we may be able to prove that any amount over $\$26$ can be produced by some combination of the two coupons. How would we know for sure that $\$329876$ could be paid out?
Our best strategy is to begin by tinkering, and then see if there are any interesting patterns that emerge. We can then try to establish through careful reasoning that the patterns we see are no fluke, that they have to continue.
So let’s start.
The first few dollar amounts one can figure out just by tinkering:
$\$27$ stock can be paid by using three $\$5$ coupons and two $\$6$ coupons.
$\$28$ stock can be paid by using two $\$5$ coupons and three $\$6$ coupons.
$\$29$ stock can be paid by using one $\$5$ coupon and four $\$6$ coupons.
$\$30$ stock can be paid by using no $\$5$ coupons and five $\$6$ coupons.
$\$31$ stock can be paid by using five $\$5$ coupons and one $\$6$ coupon.
Wait a second. Isn’t this sort of tedious? Doesn’t it seem like this approach would take us forever, and even then we would still have an infinity of numbers left to check? And weren’t we supposed to look for a pattern? There doesn’t seem to be any pattern in the $\$5$’s and $\$6$’s at all.
Well, let’s think about a $\$32$ stock. Traci knows how to purchase a $\$27$ stock—three $\$5$ coupons and two $\$6$ coupons. What coupon should she now add in order to purchase a $\$32$ stock?
Now convince yourself that Traci’s claim is valid.
Phillip says that every whole number has an even number of positive integer factors (for example, 6 is $1 \cdot 6$ and $2 \cdot 3$ so its factors are 1,6,2,3). Either explain why he is correct, or give a counterexample.
Think of any three consecutive integers and add them together. If you repeat this a few times you will notice something interesting about the sum. What is it?
If I own 12 blue socks, 16 red socks, and 18 green socks, and I reach in the dark in to my sock drawer, how many socks will I have to take out to guarantee that I have a pair that are the same color?
In the diagram below, the area of the gray rectangle is $x \cdot 1 = x$, while the area of the whole square is $x \cdot x = x^2$. Ringo says that since the gray rectangle fits inside of the square, this shows that $x$ is smaller than ${x^{\rm{2}}}$ for any positive number. What do you think of Ringo’s argument?
In the “matches” game, 30 matches are laid on a table. The first player picks up anywhere from 1 to 6 matches, the second player also picks up anywhere from 1 to 6 matches, and the two then alternate picking up 1 to 6 matches until all the matches have been picked up. The player who picks up the last match loses. Can the first player always win with clever play?
If $\lfloor k \rfloor$ represents the smallest integer greater than or equal to $k$, we can agree that $\lfloor 2 \rfloor = 2$ and $\lfloor 5.3 \rfloor = 6$. Are either of the following true?
i. $\lfloor x \rfloor + \lfloor y \rfloor = \lfloor {x + y} \rfloor$
ii. $\lfloor x \rfloor \lfloor y \rfloor = \lfloor {xy} \rfloor$
The French mathematician, Pierre Fermat (1601-1665) claimed that no one could find positive integers $n$, $m$ and $r$ such that ${n^x} + {m^x} = {r^x}$ for any integer $x$ greater than 2. But in an episode of The Simpsons, the equation ${1782^{12}} + {1841^{12}} = {1922^{12}}$ appears. What do you think?
Prove that if there are 6 people who can arbitrarily shake hands with one another, there is always a pair of persons who shake hands with the same number of people. Then prove this is true if there are $n$ people.
When asked to square a certain integer, Joshua came up with an answer of 637812, Wendy came up with 637813, and Anthwaran with 637818. Without using a calculator, determine which of them could be right.
Can one make change for a 25-ruble bill, using exactly ten smaller bills of denominations of 1, 3, or 5 rubles? (The 1, 3, 5, and 25 ruble denominations used to be standard in Russian currency.)
Imagine you are at a school that has a row of 500 lockers, all shut. Initially a student goes along the row and opens every locker. Then a second student goes along and shuts every other locker beginning with locker number 2. A third student changes the state of every third locker beginning with locker number 3—that is, if a locker is open the student shuts it, and if a locker is closed the student opens it. A fourth student changes the state of every fourth locker beginning with locker number 4.
This continues until all 500 students have followed the pattern with the 500 lockers. At the end, which lockers will be open and which will be closed?
In a certain year there were exactly four Fridays and exactly four Mondays in January. Gisele says that, given only that information, she can determine what day of the week the 20th of January must fall on in that year. Is Gisele correct?
There are 12 million people in New York City. The number of hairs on a human head never exceeds 750,000. Must there be 2 people in New York who have the same number of hairs on their head?