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        riddles >> medium >> Sum and Product
        
(Message started by: James Fingas on May 14^th, 2003, 5:59am)

Title: Sum and Product
Post by James Fingas on May 14^th, 2003, 5:59am

Quincy, Willywutang's summer student, stared blank-faced at the screen. It just didn't make any sense. The column of numbers defied explanation. "-503, -sqrt(2), 2/3, pi, 10, ..." he thought to himself. There seemed to be no pattern.

He had been sitting there for about five minutes when Willywutang walked in.

"What are you trying to do to my computer?" he asked sternly.

Quincy deflected the sharp inquiry easily--at the beginning of the summer, he would have been worried, but distracting his supervisor was now second nature to him. "What do these numbers mean? There's no pattern to them at all!"

"Oh, those numbers ..." Willywutang started. He didn't really want to explain the whole thing. At the moment, it was all very theoretical. "Yeah, they're just random. I'm doing a study on real numbers that add to 100. Those are just some pairs of numbers that add to 100. See, right below the -503 ..."

Willywutang was now looking at the screen. "Hey," he said, "you sorted them! What did you do that for?"

"I didn't know they were in some special order." Quincy replied apologetically. It would take a little more fancy footwork to defuse Willy's wrath once again. "What's this really big number over here ... 2.7x10³⁵?"

"That's just the product of all the numbers."

"And this one, under it? It looks more reasonable ..."

"That's the sum of all the numbers."

What is the sum of all the numbers in the column?

Title: Re: Sum and Product
Post by Papa Homer on May 14^th, 2003, 3:36pm

Probably some multiple of a 100 :) Like [hide]1100[/hide].

Title: Re: Sum and Product
Post by towr on May 15^th, 2003, 12:39am

Peculiar, I get an answer between about [hide]1104[/hide] and [hide]1195[/hide] but of course the right number also has to be [hide]a multiple of 100, since every pair sums to 100[/hide].

Title: Re: Sum and Product
Post by wowbagger on May 15^th, 2003, 2:14am

Strange. I found the same bounds as towr. Which is reassuring, but not really of much help.

Are you sure you got your numbers right, James? ;)

Title: Re: Sum and Product
Post by James Fingas on May 15^th, 2003, 7:47am

I thought it would take longer to find out there was no answer ... I was just amused by the question about the N numbers that sum to 20, with the smallest three summing to 5 and the largest three summing to 7, so I recast it with no solution.

Obviously you're using a different method than I used (I got ranges on the product rather than ranges on the sum).

"I think there's something wrong with that product number," said Quincy, "when did you calculate it?"

"What makes you think it's wrong?"

Quincy sketched out a brief proof, concluding: "If that were the product, there'd be no possible sum!"

"Hmm," Willywutang said, pensively, "Maybe it's old data--try calculating it again."

Sure enough, when Quincy calculated the product, he got a different answer: 1.4x10⁵⁹. If he were to recalculate the sum of the numbers, what answer would he get?

Title: Re: Sum and Product
Post by wowbagger on May 15^th, 2003, 7:54am

The sum should be [hide]1900[/hide].

Title: Re: Sum and Product
Post by James Fingas on May 15^th, 2003, 8:05am

And a follow-up question: if the miscalculated product was the sum of only the first M numbers in the list, what is M?