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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

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  • $\begingroup$ 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? $\endgroup$ – almagest Mar 21 '16 at 16:32
  • $\begingroup$ 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. $\endgroup$ – Tom Finet Mar 21 '16 at 16:35
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    $\begingroup$ 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. $\endgroup$ – Jyrki Lahtonen Mar 21 '16 at 16:37
  • $\begingroup$ @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 :-) $\endgroup$ – Jyrki Lahtonen Mar 21 '16 at 16:38
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    $\begingroup$ 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. $\endgroup$ – Tom Finet Mar 21 '16 at 16:44
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Firstly we ask ourselves: how many segments are there of length 5 cm which horizontal fit in the square? This operation is simple; 7 x 2 = 14 as 2 segments of length 5 cm fit on each horizontal line of the 7 by 7 square. Using the symmetry of the square we deduce that the number of segments length 5 cm that fit vertically is also 14, resulting in a total of 28 segments.

Secondly we must find how many segments with the vector (3, 4) -- this has a length of 5cm -- fit on the grid. This is done by counting the number of these segments that can fit on one row of the 7 by 7 square. This ends up being 4, so 4 x 3 = 12 segments (since there are 3 other rows the segments with vector (3, 4) could fit on). Next using the symmetry of the vector, it is important to realise that the segment with vector (3, 4) can be in 3 other positions. Therefore, 12 x 4 = 48 segments.

Finally, add the both answers together: 28 + 48 = 76 segments.

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