Each of 2001 coins are assigned the value 1, 2, or 3 and they are arranged in a row.
• Between any two coins assigned the value of 1, there is at least one coin.
• Between any two coins assigned the value of 2, there are at least two coins.
• Between any two coins assigned the value of 3, there are at least three coins.

Determine the maximum number of coins, assigned the value of 3, that could be in the row.

*** Adapted from a problem appearing in a Russian Mathematical Olympiad.

501 3s
500 2s
1000 1s

Looking at the 3rd condition in isolation, every 4th coin could be a 3 including the 1st and last if N is 1 mod 4.

For N coins where N is 1 mod 4, there could be a maximum of (N-1)/4 + 1 = 501 3s for N=2001.

The pattern 31213121312131213...31213
satisfies all three conditions.

Posted by Larry on 2023-05-07 07:49:18