The problem reads:
2^15 = 32768 and the sum of its digits is 3 + 2 + 7 + 6 + 8 = 26.
What is the sum of the digits of the number 2^1000?
Please do NOT post a solution as that is not remotely what I am seeking. I am just a little stuck and hoping someone can give me a non-spoilery nudge in the right direction. I have discovered a periodic
pattern in the powers of 2 and the sums of their digits, but knowing this pattern is not helping me determine what the next sum will be.
The first 6 powers of 2 are 2, 4, 8, 16, 32, 64. Their digits sum to 2, 4, 8, 7, 5, 10. If we adopt the rule that we must continue to sum the numerals until we have a single-digit result, then this becomes 2. 4. 8. 7. 5. 1. For want of a better term I am calling these numbers the "reduced" sums. This 6-element pattern of reduced sums recurs predictably through the series of powers of 2 (at least as far as 2^42).
Unfortunately, the reduced sum is not what the problem asks for. Somehow from that pattern of reduced sums, I have to figure out what the
actual sum would be. Looking at the next six powers of two, we find the sums 11, 13, 8, 7, 14, 19. Two fit the pattern without modification, but the other four must be reduced in order to fit. Hereafter, all sums fit the pattern only after reduction. But I don't see a way to predictably work backward from the reduced sum.
In general the sum of the digits tends to be larger than the number that came before; but there are plenty of local exceptions to this trend. The first seven sums that reduce to 4 are: 4, 13, 22, 31, 40, 58, 67. The first five elements in this series are promising as they increase without omission. The sixth, however, leaves a gap: 49 also reduces to 4, but it is skipped. The sums that reduce to 7 are even more perplexing: 7, 7, 25, 25, 43, 61, 61. Only four elements are distinct; three are repeated, seemingly unpredictably. However, here at least no sum is smaller than the one that came before. That's not the case with the 5's: 5, 14, 14, 41, 41, 59, 50. Huh?
Please, can someone help me stop spinning my wheels?