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        riddles >> easy >> Prime pairs
        
(Message started by: BNC on Mar 25^th, 2003, 3:09pm)

Title: Prime pairs
Post by BNC on Mar 25^th, 2003, 3:09pm

1. Can you arrange the natural numbers 1..10 into 5 pairs, such that the sums of the pairs will be 5 different prime numbers?

2. Repeat 1 for 1..50 (25 pairs, 25 different prime numbers)

Title: Re: Prime pairs
Post by Icarus on Mar 25^th, 2003, 3:36pm

For (1) there is :[hide]1+4, 2+5, 3+8, 6+7, 9+10[/hide]

Title: Re: Prime pairs
Post by NickH on Mar 25^th, 2003, 3:39pm

1. [hide]2+3=5, 1+6=7, 4+7=11, 5+8=13, 9+10=19.[/hide]
2. [hide]I'll have to come back to this one later...[/hide]

Title: Re: Prime pairs
Post by cho on Mar 25^th, 2003, 4:53pm

For no. 2: [hide] No. There are only 24 primes from 3 to 99.[/hide]

Title: Re: Prime pairs
Post by cho on Mar 26^th, 2003, 5:46am

For a slightly harder question, what is the highest number for which you could solve the riddle?

hint [hide] If I'm right, the solution for that number is quite simple at this point. Proving there are no higher solutions would be more difficult. [/hide]

Title: Re: Prime pairs
Post by mistysakura on Mar 27^th, 2003, 2:30am

on 03/25/03 at 16:53:57, cho wrote:

For no. 2: [hide] No. There are only 24 primes from 3 to 99.[/hide]

Yes, but your sums can be higher than 99.

Title: Re: Prime pairs
Post by BNC on Mar 27^th, 2003, 3:56am

on 03/27/03 at 02:30:33, mistysakura wrote:

Yes, but your sums can be higher than 99.

No, they can't. 49+50 is the highest possible pair sum.

Title: Re: Prime pairs
Post by cho on Mar 28^th, 2003, 8:41pm

For the highest possible solution, the solution is simple. [hide]Take the 10 answer and add 11+12=23. I suggest that there is no higher answer.
14: The pairs sum to 105 but the available primes only add up to 98. (Primes must be higher than the pairs because some will not be used).
16: Pairs=136; Primes=158, but you can't use both 31 and 29 in the answer (16+15=31, 14+13 is only 27).
18: Pairs=171; Primes=158. No new primes were added, so primes fall short again. At this point you see you don't have to try all possibilities, just those that add new primes.
20: Pairs=210; Primes=195
22: Pairs=253; Primes=279, but you can't use both 43 and 41.
24: Pairs=300; Primes=326, but you can't use 47,43, and 41 all together. See, the highest numbers in the solution must average at least 4 apart. (24+23=47,22+21=43,20+19=39). You can not use primes that are 2 apart unless you've left one of these higher possibilities unused.
From this point on the pairs total gradually outpaces the primes, and the only time the primes do some catching up is when you have consecutive primes, but then you can't use them both anyway. By the time you reach 50, the pairs are over 200 points ahead of the primes, and since primes become more and more widely separated, I don't believe they could ever catch up. [/hide]