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Numbers answer

A few days ago I asked what the next pair of numbers in this series is:

49, 1
50, 10
53½, 21½
55, 25
60½, 35½
62, 38

pwilkinson has posted the correct answer:
70,50

The diference between the squares of each pair of numbers is 2400:
492 - 12 = 2401 - 1 = 2400 (48*50)
502 - 102 = 2500 - 100 = 2400 (40*60)
53½2 - 21½2 = 2862¼ - 462¼ = 2400 (32*75)
552 - 252 = 3025 - 625 = 2400 (30*80)
60½2 - 35½2 = 3660¼ - 1260¼ = 2400 (25*96)
622 - 382 = 3844 - 1444 = 2400 (24*100)
702 - 502 = 4900 - 2500 = 2400 (20*60120)

And so on up to 1200½2 - 1199½2 = 1441200¼ - 1438800¼ = 2400 (1*2400).

Comments

( 8 comments — Leave a comment )
slovobooks
Feb. 27th, 2007 08:09 am (UTC)
I was definitely on the wrong track, so!
ahousekeeper
Feb. 27th, 2007 08:26 am (UTC)
Me too :)
pgmcc
Feb. 27th, 2007 10:17 am (UTC)
Now we see the dangers of long waits at airports.
blonde222
Feb. 27th, 2007 10:55 am (UTC)
I don't efven know how you would begin to go about working out the answer to that.
pwilkinson
Feb. 27th, 2007 07:56 pm (UTC)
I'll admit I went slightly the long way round on this one.

I started (after a few minutes staring at the numbers) by spotting that the sums of the pairs of numbers were divisible by several numbers even though the individual numbers didn't (for instance, 53½ and 21½, besides not even being whole numbers, are prime when multiplied by 2 - but they add together to 75, which is divisible by 3, 5, 15 and 25).

Something else was still obviously needed and the first thing I now looked at was what happened if I subtracted the second number in each pair from the first - and I saw an obvious pattern. Of course, it does help to have spent much of one's life looking for or at somewhat similar patterns.
hells_librarian
Feb. 27th, 2007 11:54 am (UTC)
I like these series questions, even if I've never figured one out. I love that "Oh, that's nifty" moment.
cassave
Feb. 27th, 2007 12:57 pm (UTC)
# 70² - 50² = 4900 - 2500 = 2400 (20*60)

Shouldn't the bolded numbers be "(20*120)"? ;-)
nwhyte
Feb. 27th, 2007 08:30 pm (UTC)
Er, yes.
( 8 comments — Leave a comment )

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