Stage Four
Handshakes
A new school, Smartsville High School, is opened and there are 100 students in Year 8.
Students meet each other by greeting each person with a handshake.
If every student meets every other student, how many handshakes will there be?
Hint: using a smaller number of people create a table and look for a pattern.
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Number of handshakes |
You may like to act it out!
Making a table up:
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I worked these values out myself using diagrams like these:
Each line represents a handshake.
What is the pattern here? Notice that for any number of people if you add the number of people to the number of handshakes, the result is equal to the number of hand shakes for the next higher number of people...
2+1 =3, 3+3=6, 4+6=10, 5+10=15, 6+15=21, etc.
A little examination resulted in finding a pattern: 1+2+3+4+5...
up to the number 1 less than the total number of people.
Another way to think about a solution is this:
Think about just one person to start with. If they shake hands with everyone, they will shake hands 99 times. They can now be removed from the group.
Then the next person has 98 people to shake hands with. The next has 97.
Right down to the final situation where there are only 2 people
left, resulting in 1 final handshake.
So, either way, what will this equal? Work it out... 99+98+97+96+ etc. all the way down to 1. Get out the calculator!
A correct answer came in from students at Ashford Central School: there would be 4950 handshakes.