This is a downs only sudoku crossnumber. All answers are created from the digits 1 through 6. Per sudoku rules, each digit 1-6 appears exactly once in each row, column, and section, I.e. all sudoku rules apply. Only the downs are clued but due to the sudoku constraint, you needn’t solve all the clues. Finally, this is clued using math facts about the numbers (with one exception).
侍 SamuRai
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The sum of the first k even numbers is k^2+k. This does force some interesting sub-patterns but not the one I suggested. For example, when k is 2^n-1 (all 1s in binary) the sum is 4^n-2^n which is a 1 followed by n-1 zeros then a 1 followed by n zeros. For those interested, here is a list of the first 50 terms of the summation of even numbers in binary.
I don't know if 19D is actually a property of triangular numbers or if it's just the Strong Law of Small Numbers... But you have me really curious. For every triangular number I've checked so far, it's either a single string of 1s or alternating digits. (The sum that you described is just double a triangle)
Number of unique orderings four people can finish in a race, disallowing ties.
22D
Consider a 3x3 grid of squares with their NE-SW diagonals drawn. This is the number of ways to move from the NE corner to the SW corner, each time moving south, west, or southwest. spoiler