Riddles

[Guus]

Yeppers, you got it

[Davecom3]

A man takes turns, first drawing a horizontal line and then drawing a vertical line on a sheet of paper and then back to horizontal. It takes him 4 lines to draw one rectangle. If he continues this pattern(one vertical line after every horizontal line), how many vertical lines does it take him to draw 1 million rectangles, assuming he's using only one side of a single flat sheet of paper?

[gamecreator]

1001? (1 plus square root of 1 million)

[Davecom3]

nope, I assure you it's considerably less.

[Patdragon]

that would assume all the rectangles are the same size...(i think), I think ive heard this one before and its something like 31 but i can't remember how it was done exactly.

[Davecom3]

That's closer, but still definitely off. And you're on the right track, Patdragon.

[gamecreator]

Oh, you mean not only the smallest ones, but rectangles composed of other rectangles.
Then It is not possible to get exactly 1 million, but after 46 vertical lines he would have 1071225 rectangles.

[Davecom3]

But are you sure that's the least number?

[gamecreator]

After drawing 45 lines he would have only 980100 rectangles, so yes.

[Davecom3]

Try thinking again. You are missing something.

[Patdragon]

half that then- 23 as you only need vertical lines?

[Davecom3]

No, he's missing something, but it will probably go to him. He's calculating using vertical lines. But, by doing so, he's missing half the puzzle.

[Patdragon]

fair noth i'm stumped on it. tho my riddle mind says if he doesn't take his pen off the paper he does it in 4 or 5 (first line he draws after first 4) as he could draw one continuous line making rectangles

[Davecom3]

Horizontal lines first, vertical lines last. It takes 91 lines to achieve 1 million rectangles. Thus, he completes the million with a horizontal line. The first part of the puzzle involves squares of triangular numbers. The second involves what happens by adding a single line. Anyways, giving it to gamecreator.

[gamecreator]

The text is ambiguous. I thought the step is finished only after a vertical line is drawn. But there is nothing in the text, that supports or contradicts this. So the man would have to draw 46 horizontal and 45 vertical lines. He would get 1024650 rectangles (however, still not exactly the million).

[Davecom3]

If he's drawn 1,024,650 rectangles, he's also drawn 1 million rectangles. This is like how all months have 28 days. If he's drawn 1 million rectangles, having completed these with a horizontal line, even if afterwards he draws a vertical line, he didn't need to draw that vertical line to achieve that last bit of the million. But, in any case, your turn.