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### #31

Posted 18 March 2008 - 08:30 PM

### #32

Posted 19 March 2008 - 12:38 AM

I'm not sure how I missed that, since I had in front of me several sets of "sage-making" special years with consecutive evens (They're rare, however. The only sets under 1000000 are 7442,7443,7444; 20402,20403,20404; 243602,243603,243604; 647522,647523,647524)It is a good reasoning, but it is not always true. ...

So, still we can have two consecutive even numbers which fulfill the special years of Lyrians ... which means Lyrians may die.

So, our logic would hold true for avoiding three consecutive evens, since either p or q would have to repeat the 2, and there wouldn't be a corresponding q or p to satisfy the equation, but that only prevents you from having six consecutive special years. My gut feeling is that it's still impossible, but I won't be able to prove that programatically. In case it helps anyone else to see a pattern, there's a good-sized list of sequences of three special years in the spoiler (most of the sets under 1000000).

**Edited by Duh Puck, 19 March 2008 - 12:43 AM.**

### #33

Posted 19 March 2008 - 11:22 AM

That is pretty correct. Actually, the Lyrians dies ... that is, we can have four consecutive special years. I will wait in case someone comes up with the solution. Otherwise, I will post the solution tomorrow.So, our logic would hold true for avoiding three consecutive evens, since either p or q would have to repeat the 2, and there wouldn't be a corresponding q or p to satisfy the equation, but that only prevents you from having six consecutive special years.

### #34

Posted 19 March 2008 - 02:22 PM

Well, shoot. In that case I'll give it one more run tonight using significantly improved algorithms for testing primality (the same ones used for RSA encryption techniques). I just downloaded the code and tested and it turns out that the increase in time for generating a list of prime numbers is almost linear to the number of elements, while my previous method was logarithmic, which caused me to hit barriers much sooner. I just generated primes up to 1,000,000 in about one minute, meaning I should be able to reach about 500,000,000 overnight. Also, I think I can wrap the special age test in the same loop I'm using to generate the primes, which means I won't have to wait to verify output. I can't wait to get home and try it!That is pretty correct. Actually, the Lyrians dies ... that is, we can have four consecutive special years. I will wait in case someone comes up with the solution. Otherwise, I will post the solution tomorrow.

My prediction is that if Lyrians die before the age of 500,000,000, I'll tell you by tomorrow. If it's bigger than that, hey, they might as well be immortal.

### #35

Posted 19 March 2008 - 02:38 PM

Well, shoot. In that case I'll give it one more run tonight using significantly improved algorithms for testing primality (the same ones used for RSA encryption techniques). I just downloaded the code and tested and it turns out that the increase in time for generating a list of prime numbers is almost linear to the number of elements, while my previous method was logarithmic, which caused me to hit barriers much sooner. I just generated primes up to 1,000,000 in about one minute, meaning I should be able to reach about 500,000,000 overnight. Also, I think I can wrap the special age test in the same loop I'm using to generate the primes, which means I won't have to wait to verify output. I can't wait to get home and try it!

My prediction is that if Lyrians die before the age of 500,000,000, I'll tell you by tomorrow. If it's bigger than that, hey, they might as well be immortal.

ok ... I am sure you will get it. Just one comment, the number is much bigger ... I am not sure about your algorithm, but if possible try to start it with a big number in order to cut some iterations of lower value. I have put the range as a hint in the spoiler.

### #36

Posted 19 March 2008 - 08:10 PM

Well, by doing prime factorization and ruling out a lot of non-possibilities, I got it faster, but not that much faster! It looks like I'll get up through about 2 billion by tomorrow, but unfortunately I can't start at a higher number, since I need an in-memory list of primes to efficiently do factorization. I think I'll eventually get it, but it would be nice to see a more elegant approach.ok ... I am sure you will get it. Just one comment, the number is much bigger ....

### #37

Posted 21 March 2008 - 04:14 PM

### #38

Posted 21 March 2008 - 04:52 PM

### #39

Posted 21 March 2008 - 04:56 PM

Woohoo! I realized some significant changes I could make to optimize the program, after which it solved the problem in 11 mins.

Spoiler for Program output

Here we go .... the Grand Slam title of the Lyra III goes to ...

**Duh Puck**. Well done.

**Edited by brhan, 21 March 2008 - 04:58 PM.**

### #40

Posted 21 March 2008 - 04:57 PM

Agreed. I'm curious how you came up with the idea for this one. I'm still assuming there's a swank mathematical proof for arriving at the result in a much more efficient manner than brute force, but there must have been an AHA! moment when you saw or discovered some pattern and realized it would make for a cool puzzle.brhan - this is a very cool original puzzle.

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