# Math Problem: Forty-nine points

49 points are marked on a sheet of paper in a square. Adjacent points horizontally or vertically are separated by exactly 1 centimetre.

How many straight lines of length 5 centimetres can be drawn between points in the design?

I have attempted to solve this problem taking into account the lines going through 6 points and the 5 cm diagonal lines with height 4 cm and width 3 cm. However, these approaches have been unsuccessful. The answer is 76 lines, but I cannot see how to reach this value. How is this problem solved and am I missing anything?

Thanks

• I am not clear how the points are drawn. Suppose the first is at the origin $x=y=0$. What are the possible positions for the second? Also what do you mean by "adjacent" points? – almagest Mar 21 '16 at 16:32
• The points are drawn on a piece of paper in a 7 by 7 square. Adjacent points are points that are next to each other either vertically or horizontally separated by 1 cm exactly. – Tom Finet Mar 21 '16 at 16:35
• Your beginning is correct (+1). To make progress answer the following: How many such horizontal line segments are there? The number of vertical line segments is the same by symmety. How many represent the vector $(3,4)$? Again by symmetry the vectors $(3,-4)$, $(4,3)$ and $(4,-3)$ occur as often as $(3,4)$. You only need to get your hands dirty twice - the rest follows from symmetry. – Jyrki Lahtonen Mar 21 '16 at 16:37
• @almagest: I'm fairly sure that we should imagine a 7x7 square grid, with the small squares having sides of length $1$ cm. At least that leads to the answer 76 :-) – Jyrki Lahtonen Mar 21 '16 at 16:38
• There are 14 horizontal segments so due to the symmetry 14 vertical segments. 12 segments represent the vector (3, 4). So 12 * 4 is 48 segments in total. Therefore adding both together we get 76 segments. Thanks for working me through it. – Tom Finet Mar 21 '16 at 16:44