Q: This challenge appeared in a puzzle column in the Woman's Home Companion in January 1913, exactly 100 years ago. Draw a square that is four boxes by four boxes per side, containing altogether 16 small boxes and 18 lines (across, down and diagonal). There are 10 ways to have four boxes in a line — four horizontal rows, four vertical columns, plus the two long diagonals. There are also eight other shorter diagonals of two or three squares each. The object is to place markers in 10 of the boxes so that as many of the lines as possible have either two or four markers. What is the maximum number of lines that can have either two or four markers, and how do you do it?Sorry, I needed my full 8 hours of sleep so wasn't able to post earlier, but I'm awake now. No doubt people are going to find this a tricky puzzle, so I added a diagram to help out. Apart from reflections or rotations, I've found a single solution that maximizes the number of lines. Please don't mention how many lines are involved until after the deadline. For those that are mathematically inclined, there are 8008 ways to place 10 markers into 16 squares and, accounting for symmetry, it shouldn't be too hard to brute force the answer. :)
Edit: My hints above were to the number of lines being 16. Being asleep for 8 hours leaves 16 waking hours. The group "No Doubt" has a song called "Sixteen" on their album "Tragic Kingdom". Also the reference to the number of possible arrangements (8008) hinted at 8 horizontal/vertical lines and 8 diagonal lines. Finally, in a post I mentioned the Beatles' song "Taxman" which was parodied by Weird Al as "Pac Man"... I think the answer looks like Pac Man eating a dot.