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Dodge the 9s
Posted: Wed Aug 03, 2016 8:53 pm
by Bob78164
This puzzle comes from the New York Times. What is the longest arithmetic progression of positive integers you can construct in which none of the numbers include the digit 9. For example, one such progression is 1, 8, 15, 22, which has length 4.
The longest such progression I've been able to devise has length 72. Can anyone do better? --Bob
Re: Dodge the 9s
Posted: Thu Aug 04, 2016 10:05 am
by Pastor Fireball
Nope. That's the best I can do. Increase 1 by 125 a total 71 times to hit a maximum of 8876.
Re: Dodge the 9s
Posted: Thu Aug 04, 2016 10:18 am
by Pastor Fireball
And it appears that you can also get 72 numbers when you increase by 125,125. So the limit of such an arithmetic progression seems to be 72, and it will presumably work for any n-digit progression where n is divisible by 3 and the digits are 125 repeating.
Re: Dodge the 9s
Posted: Thu Aug 04, 2016 10:45 am
by Bob78164
Pastor Fireball wrote:And it appears that you can also get 72 numbers when you increase by 125,125. So the limit of such an arithmetic progression seems to be 72, and it will presumably work for any n-digit progression where n is divisible by 3 and the digits are 125 repeating.
I don't think this latter one works. The 33rd entry is 4,004,001, so the 34th entry is 4,129,126, which has a 9.
But increasing by 1,250,125 should work. --Bob