从 a 到 b
Getting from a to b
题目详情
We’re making a tower!
Initially, nobody thought out the huge energy needed. Even quite useful approaches looked suspect, adding tons of thought heretofore even beginning.
There have been some negatives along the way, but also some positives. But we’ve powered through it, one way or another, and things are looking up.
As you can see, rows 1 through 7 have already been built. (And we’ve got the next 9 figured out as well.)
However… I found this one piece near the bottom of the bag, I’m having trouble figuring out where it will go. Can you determine the first (lowest) row where it would make sense to place it?
解析
Original Explanation
OK, so it took a little longer than expected, and we even had to go find a step stool, but as you can see (above) we eventually finished the tower :-)
Astute puzzlers noted that the sentence that began “Initially, “, and which then proceeded to sound like chatbot slurry, was itself a major clue. The initial letters from the words that followed spell out “n to the n equals a to the b”.
And so: the nth level of the tower displays the number of ways to rewrite nn in the form ab, where a and b can be positive or negative integers. (As alluded to in the third paragraph.)
The least integer n for which nn permits exactly 2023 such re-writings is 22 × 316 × 516 = 26,273,633,422,851,562,500.
The casework involved in solving for the least such n is fairly interesting; we’d love to share it but we’ve just been told we need to disassemble the tower tout suite to clear away table space for the research team’s daily game of Campaign for North Africa. Gotta go!
Congrats to this month’s solvers!