骑士动作 4
Knight Moves 4
题目详情
A knight was initially located in a square labeled 1. It then proceeded to make a series of moves, never re-visiting a square, and labeled the visited squares in order. When the knight was finished, the labeled squares in each region of connected squares1 had the same sum.
A short while later, many of the labels were erased. The remaining labels can be seen above.
Complete the grid by re-entering the missing labels.
The answer to this puzzle is the sum of the squares of the largest label in each row of the completed grid, as in the example.
[1]: E.g. the 14 and 33 are in different regions.
解析
Original Explanation
The above image shows the reconstructed path of the knight. The move count of 50 (and region sum of 75) can be deduced using three observations:
- the sum of all numbers must be a triangular number,
- the sum of all numbers must be divisible by the number of regions (17), and
- the regions of size 2 must have an odd sum by parity arguments of knight moves.
Once this is known the path of the knight can be deduced by hand or with the aid of a program (we saw both solution methods). The sum of the squares of the maxima of each row comes to 14820.
Congrats to this month’s solvers who successfully deduced the knight’s path!