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## Tiny Little Stairs

Math brain teasers require computations to solve.

 Puzzle ID: #11242 Fun: (2.72) Difficulty: (1.83) Category: Math Submitted By: jimbo Corrected By: shenqiang

Jimbo Gulliver was planning an intergalactic space trip to the planet Lilliputinyweesmallminutania which had a very tiny moon named Tinylittliddybiddy... well never mind what it was called. Here dwelt the smallest people in the universe.
In order to give them access to his space ship doorway, which was 3 metres off the ground, he decided to build a set of stairs that measured 4 metres long by 3 metres high.
Just as he was about to leave, the first officer on the ship, Lieutenant Sarogahtyp said "Jimbo, when we get there are we going to give them the red carpet treatment?"
"Good idea", declared Jimbo.
Now Jimbo had the latest autocarpetachometer thingy that could fasten a carpet to a staircase without wasting any carpet whatsoever on the corners and bends. "I wonder how much red carpet I'll need?" enquired Jimbo.
"Look", said first officer Sarogahtyp, "There are thousands and thousands of steps. The steps are so tiny and close together that it almost looks like a ramp. Why it's almost a 3:4:5 triangle. Take 6 metres and you'll have some left over.
Jimbo Gulliver took his first Lieutenants advice but alas, when he got there, there was not enough carpet to carpet the stairs! Why?

### Answer

It will take exactly seven metres to carpet the stairs no matter how many there are. The horizontal steps must add up to 4 metres and the vertical risers must add up to 3 metres. Total 7 metres, even if there are an infinite number of stairs. This appears contradictory to our sense that it flattens out to become like the hypotenuse of a right angled triangle. A quick sketch with any number of stairs should convince you.
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