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'Crossing the Town Square' printed from

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Imagine the central square in a big city and its paved with large square tiles. It may be rectangular rather than square! You are going to go straight from one corner, diagonally across to the other corner. You may be walking, cycling, skate boarding or using roller blades. Which ever way you travel you will need to go absolutely straight from corner to corner.

You see in the picture above showing a very, very small example (a $4$ by $3$ rectangle). The blue line of travel goes through six of the square tiles. Maybe there are other small rectangles other than this one that crosses $6$ tiles.

Your challenge is to find what different sizes of rectangles would mean you travelled across $10$ tiles. 

Your extra challenge is to find a set of answers for $12$ tiles being crossed.

Can you find a generalization/pattern/system from the ones you've done that would enable you to find solutions more easy to other numbers of tiles being crossed.