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Dinner party clinks

October 21, 2012

A few weeks ago I asked:

At a dinner party for 10, each person ‘clinks’ their glass once with each other person. How many clinks?”

I’ll just leave some space here for you to think about it…

 

 

 

 

 

 

 

 

 

 

 

 

 

Got any ideas? Maybe you know the problem already. There is more than one way to think about this one (as there often is). Here’s two…

1. Purposely double count

Well, each of the 10 people ‘clink’ with the other 9, so that’s 10 x 9 = 90 clinks, right?

Not quite – we’ve counted each clink twice.

For example, we counted when person 1 clinked with person 2, then we counted again when person 2 clinked with person 1.

So, there were 10 x 9 / 2 = 45 clinks.
Similarly, for 12 people, there would be 12 x 11 / 2 = 66 clinks.
More generally, for n people, there would be n(n-1)/2 clinks.

2. Count as we go

Person 1 clinks with 9 others.
Person 2 clinks with 8 people, not counting the already counted clink with person 1.
Person 3 clinks with 7 people, not counting the clinks with persons 1 and 2.
Person 4 clinks with 6 people, not counting the clinks with persons 1 to 3.
Person 5 clinks with 5 people, not counting the clinks with persons 1 to 4.
Person 6 clinks with 4 people, not counting the clinks with persons 1 to 5.
Person 7 clinks with 3 people, not counting the clinks with persons 1 to 6.
Person 8 clinks with 2 people, not counting the clinks with persons 1 to 7.
Person 9 clinks with 1 people, not counting the clinks with persons 1 to 8.
Person 10 has had all their clinks counted already!

So the number of clinks is 9+8+7+6+5+4+3+2+1 = 45

7 year old Gauss’s formula

See my earlier post here for the relationship between the above two methods.

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