Question
Dear Mitch,
Like just about everyone else whose been writing in (or at least whose letters you've been responding to), I was wondering if you could keep those magic tips coming for the SAT, at least until the first week of October, when we have to take the PSAT. My girlfriend and I find them really amazingly helpful. She prints them out and we go through them together. So if we do as well as we're hoping to, we'll definitely owe you forever!
Sincerely,
Connor T.
Roanoke,VA
Answer
Dear Connor,
My pleasure.
There is a question that shows up on about a third of the SAT's, and is rather simple to solve correctly, but FIRST it has to be recognized. And that's the critical part, the recognition of the concept, as it can be disguised in many forms.
For a long time it was referred to simply as 'the handshake problem', mainly because the fact pattern almost always involved people shaking hands at some form of get-together.
Now, though, it shows up in a variety of ways, and after I go through the basic concept I will try to list the most common ways it can be disguised.
Question:
Seven people attend a meeting, and they each shake hands with each other once.
How many handshakes take place at that meeting?
(And since it can show up as either a multiple choice or a 'student produced response', I will not subject you to a menu of choices.) Sixty seconds, ready go!
Answer?
VERY IMPORTANT: WRITE YOUR ANSWER DOWN, IN INK, AND CIRCLE IT. This way, if you get it correct you will not have a hard time convincing yourself that you did in fact get it, as the writing will be there in front of you. However, in the unlikely event you do not arrive at the answer that one would consider ideal, but have a bunch of different smudges and half-completed calculations that could be interpreted any way you please, then you will surely learn the disadvantage of not committing to an answer; at this stage (the studying part prior to exam), is far more useful to have an incorrect response written down than no response. The incorrect one will serve you well as you improve and watch each incremental change. It is highly rewarding to actually see improvement in your work. So: WRITE SOMETHING DOWN and circle it, even if you are sure it is not correct. You cannot learn much until you know what you know and – more importantly – know what you do not really know. So if you circle your inked answer, and it is not correct, you cannot fool yourself into thinking you 'sort of almost kind of know it mostly'....
Answer written down? If not, take another sixty seconds and try again.
Answer written down and circled?
Okay, the answer is....
The winning score in a typical game of table tennis, or the best number for your cards to add up to in blackjack. It is also the only number less than 25 that is a multiple of 3 and 7.
Why didn't I just come out and say the answer? To protect you, in the event of your eye accidentally roaming too far down the screen before you actually tried to work it out yourself. One last time, the answer is the number between 20 and 22 that rhymes with fwenty-son.
Okay, first, here's the easiest method:
You draw a sequence of horizontal dashes on your scrap paper, one dash (or blank) for each person attending the meeting and involved with the handshaking; here it is seven.
So: _ _ _ _ _ _ _
(Each blank represents a person).
And you start on the left and ask yourself how many people will that first guy shake hands with? (For the sake of brevity, rather than writing 'he or she', I tend to alternate between the two genders, and today the blank at the meeting is a 'he', or 'guy').
Well, since he does not shake hands with himself, the answer is 6. Put a 6 in that blank.
Next: How many hands can the second blank-guy then shake hands with? NOT 6, because he has already shaken hands with one of the people, guy-blank #1, so the answer is 5. Write a five in the second blank. The third blank? 4, because two people have already shaken his hand. Then, following the pattern, 3, then 2, then 1, and zero – because the last guy has already had handshakes with everyone else attending the meeting.
You then add up the numbers in the blanks.
NOTE: I've probably had the pleasure of watching students do this type of question a thousand times over the years, and almost every one of them who gets this far in his/her work then does something that strikes me as unnecessarily challenging: They do the adding up from left to right. DO NOT DO THAT. Why? Because you are spending the majority of your adding time working with relatively large numbers instead of relatively small ones. That increases your chance of making an error. Start with the zero, then the 1, the 2, the 3, and the 4 add up to 10 (as you may know from the set up of bowling pins), and to this you add 5 and get 15, and then add 6 and get 21. In this case the numbers are all so small it hardly matters what order you do your work, but in plenty of problems it can make a world of difference.
NOTE: THE CHANCES OF COMING ACROSS THE WORD 'HANDSHAKE' ON THE PSAT OR SAT ARE NOT HIGH. HOWEVER, THE CHANCES OF COMING ACROSS A PROBLEM LIKE THIS ARE QUITE HIGH. HOW? WHAT? HOW?
The key ingredient to this type of problem is two individuals coming together to experience one event. Two people meet and shake hands: 2 people, one handshake.
So where else could you find such an oddity???
Hugs.
Fights (two people get together and have one argument).
Kisses.
Clinking of wine glasses when someone makes a toast: two glasses, one clink.
And, a few years ago, the SAT had this: A baseball league with twenty-one teams... how many different arrangements of team against team could there be if each team played every other team once? DRAW 21 blanks and begin!
Hope this helps,
Mitch
P.S. And don't forget to give your girlfriend a big handshake.
P.P. S. And, my friend, you REALLY OWE ME BIG TIME!
(just kidding)