GeekDad Puzzle of the Week: Landscape Genuis – Solution

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CobblestonepavingstoneCobblestonepavingstoneCongratulations to landscape foreman Jacob Storey who gets a $50 gift code to ThinkGeek. Thanks to all who submitted solutions. Check out the solution after the jump and your $10 off your purchase of $30 or more at ThinkGeek.

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Special thanks to ThinkGeek for providing our prizes!

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Sometimes even a genius must do a bit of mundane DIY home improvement from time to time. Some spare days in between projects finds you outside laying paving stones for your new back patio. It’s coming along nicely and your stones have all been placed with your usual care and precision but there’s a problem. You’ve got left seven empty spaces in various spots and a limited amount of stone remaining.
Your middle son comes out back to find you with furrowed brow and suicidal thoughts.

"What’s wrong dad?"

"Well, son, I’ve got left over spaces in my stonework here. Each space will fit an isosceles right triangle. I’ve also discovered that no two triangles are congruent and all I’ve got here is one square piece of stone. How am I going to use it all exactly and cut seven of these triangles from it to finish our patio?"

Your son looks at the square piece of stone for a bit and says:
"I’ve got it dad! Let me borrow that pencil and I’ll show you how to cut it."

How does your son diagram cutting the stone so that none of it is left over but provides seven isosceles right triangles, no two of which are congruent?

Solution

Amazingly, your son draws his lines and the triangles fit the spaces in your stonework perfectly! There are actually two valid ways to tile seven isosceles right triangles in a square:

Landscape_solution1_2Landscape_solution1_2

Landscape_solution2Landscape_solution2

A good list of references and analysis of the math involved can be found at www.squaring.net/tri/twt.html. (But some of you with Google hardwired into your grey matter knew that already, didn’t you?)

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