A Conversation for Basic Sorting Algorithms

½(m×(m -1))

Post 1

Recumbentman

Surprising how often this number turns up. It is also the number of clinks heard when m people clink glasses.


½(m×(m -1))

Post 2

Sam

I'm not sure I fully understand this but I think you may have given a mathematical proof to that which I've often wondered about (actually, it's my girlfriend who always asks the question) - if, say, there are 10 of us around the table and we all raise our glasses, toast each other's health and clink each glass, how many clinks will be heard in total? Does this equation solve the problem?

smiley - smiley


½(m×(m -1))

Post 3

The Researcher formally known as Dr St Justin

Indeed it does! So for 10 of you, there would be ½(10×(10-1)) = ½(10×9) = ½(90) = 45 clinks! Although some of them may be simultaneous... smiley - winkeye

Thinking about it in a slightly different way, you could at each stage have clinks - 5 pairs of people clinking glasses. Each person needs to clink 9 other peoples glasses, so we will need 9 stages. Therefore, there should be 5 clinks each time × 9 stages = 45 clinks! smiley - ok


½(m×(m -1))

Post 4

Sam

Dr J (and Recumbantman) this is brilliant! For a maths ignoramus like me it's a bit of a mindblower. Thank you. smiley - smiley


½(m×(m -1))

Post 5

The Researcher formally known as Dr St Justin

No probs mate! We aim to please...


½(m×(m -1))

Post 6

Recumbentman

You can also picture it this way: each person (m of them) clinks with everyone else (m-1 of them). Each of these contributions is half a clink - or if 'half a clink' doesn't bring up a visual/audio image, say 'it takes two to make a clink'.

If this doesn't work the magic, don't be upset. Was it Von Neumann who said 'you don't understand maths, you just get used to it'?

When m=1 there are no clinks. (Just tried it.)


½(m×(m -1))

Post 7

Sam

Excellent.smiley - smiley


½(m*(m -1))

Post 8

Sea Change

This clink formula only works if you don't have a doofus like me in the mix. Invariably I miss some and doubleclink others.smiley - winkeyesmiley - ale


½(m*(m -1))

Post 9

Gnomon - time to move on

What is the sound of one glass clinking? smiley - smiley

The reason this particular formula (m/2)(m-1) comes up so often is that it is the sum of integers 1 + 2 + 3 + 4 ... + m. Or are we talking mathematical tautology here?


½(m*(m -1))

Post 10

Recumbentman

According to the mighty Wittgenstein (A963579) all of maths is tautology.


½(m*(m -1))

Post 11

Old Hairy

The chinking glasses theorem looks worthy of demonstration by experiment. My attempts to do this for numbers on the order of a dozen failed, because (Hic, beg your pardon) the results cannot be recollected by any of the participants. Perhaps we demonstrated that maths is a tautology - we all got fairly tight. We were all wearing glasses (see my PS), and are now recumbent in a recovery room, relishing the rest.


½(m*(m -1))

Post 12

Recumbentman

Can you clink a pair of glasses while wearing them? I did once when they were in my pocket, but they're mended now.


½(m*(m -1))

Post 13

Old Hairy

Hello Recumbent.

Happy to meet you.

"Can you ... I once did..." sounds rhetorical.

It can be done, but involves pain, both physical and economic. I suggest it is of paramount importance that this be avoided.

OH


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