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When Louis arrived at his apartment in Europe he found a note pinned to the wall in his study. The note was pinned on a 10 Euro bill. It was from his yankee roommate Jack, who always liked to play games.

The note said:

“*This 10 Euro bill is yours if you confirm for me that the dimensions of the empty box on the floor are 15Long X 11Wide X 15High in inches—yes inches!

The only thing you can use to measure the box is a flat wooden strip sitting next to the box, which is 35 inches long. Using anything in this room only, you can make only ONE mark on this flat wooden strip. There is a small pencil next to the strip. For measurements, you can only contact the strip and the box. Your measurement must be within +/- 0.05 inches.

If you can’t do it leave a 10$ bill for me buddy”.*

Imperial units, damn, thought Louis. But he couldn’t let Jack win. So he set about to measure the box.

He made some quick observations. The box was a normal cardboard box (like ones you get from AMAZON.COM). It had no top or bottom. So only 4 sides. Clearly two sides looked like a square and could be 15X15, he concluded. The wooden strip looked like a Ruler with no marking on it whatsoever. Could not bend it either. There was not much else in the room. Some frames on the wall, a study table with an old Coca Cola can, a printer with A4 paper and pens and pencils.

Then he pulled out a blank paper from the printer and wrote or sketched something. He took out his iPhone 6 plus and checked a couple of things. Then he smiled.

Using something, he made one mark on the wooded strip with the pencil. He took the box and made his first measurement. “YES” he shouted. Then he made some additional checks using the box and the ruler.

He wrote all the procedure down for Jack. He pocketed the 10 Euros and headed to the bar. “I think I deserve a beer”
enter image description here

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  • $\begingroup$ I was sure an A4 sheet was 11", it's close though lol. $\endgroup$ – BreakingMyself Jul 17 '17 at 13:30
  • $\begingroup$ "One mark" is ambiguous. Can the one mark consist e.g. of a pencil stroke running along the length of the ruler from one place to another? :-) (In the absence of further guidance I shall assume that "one mark" means we get to choose one distance from one end and mark just that.) $\endgroup$ – Gareth McCaughan Jul 17 '17 at 13:42
  • $\begingroup$ Gareth. Yes one distance from either end of the ruler $\endgroup$ – DEEM Jul 17 '17 at 13:43
  • $\begingroup$ Incidentally, some of the results you find if you ask Google about the circumference of a Coke can are a bit startling and probably NSFW. $\endgroup$ – Gareth McCaughan Jul 17 '17 at 13:45
  • $\begingroup$ To clarify the mark further; it's a single point, not a line or anything else? (e.g. I couldn't draw a line from 15 inches to 24 inches along the wood, leaving one end for each side) $\endgroup$ – Samthere Jul 18 '17 at 15:15
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This updated method uses the following:

The 10 Euro banknote is 5.00 in long (127 mm = 5 in as 25.4 mm = 1 in ) and he uses the note to make a mark 5 inches from one end of the wooden strip so he can measure 5 or 30 inches.

And he will use Pythagoras' Theorem later below with values $\sqrt{15^2 + 26^2} = \sqrt{901} \sim 30.02$

Then since it a cardboard box with no top or bottom

He folds it flat and notes it makes a rectangle like below
enter image description here
He measures both diagonals using the 30 in length of the strip and these are correct to 0.05 in.

To measure the 15 in heights, he lines up the wooden strip along the left edge and then rotates the strip a half-turn - using Penguino's tilting idea - along the top edge until it is flat against the reverse edge. The bottom of the reverse edge should be at the 30 in mark. He then repeats along the right side. This measure the 15 inch heights.

He can measure the 15 in long sides first, before flattening the box, by lining up the wooden strip along one face - say the left face of the box - and then rotate a quarter-turn along the top left edge, until it lines up along the top-edge of the front face. The 30 in mark should be aligned with the right side of the front face.

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  • $\begingroup$ Tom, so he gets 26. How does he know it is 15 and 11 exactly? If he measures 15 with the 5 inch measure would he not have to make pencil marks on the box? $\endgroup$ – DEEM Jul 17 '17 at 14:12
  • $\begingroup$ Deepak, I'm afraid this method uses pencil marks on the box to measure the 15 in sides, but he will know the 11 in side is correct as (15 + 11 = 26) and 26 is one side of the 30, 15 right-angle triangle. $\endgroup$ – Tom Jul 17 '17 at 14:17
  • $\begingroup$ @Tom What if you laid the ruler on the diagonal, then laid the Euro banknote on the part of the ruler sticking out past the corner. It should just reach the end, measuring 30 inches without making a mark. Next, make a mark on the ruler corresponding to the distance between the top crease and the bottom crease (on the part of the box "underneath" your drawing). This should be 4". 4+26+5=35. $\endgroup$ – Hugh Meyers Jul 17 '17 at 19:44
  • $\begingroup$ Hugh, irel tbbq! The 4 inch mark (could draw the reverse edge as a dashed line) may work. I wonder if allowed to measure the 30 inch with the banknote extending the diagonal - this is thinking outside the box! $\endgroup$ – Tom Jul 17 '17 at 20:17
  • $\begingroup$ @Tom, I think you would be able to ascertain whether a side is 15 without using pencil marks on the box by using the 5 inch measurement, then using the euro note where the 5 inch measurement ends, and then the 5 inch measurement again on the other side of the euro note. $\endgroup$ – Green Jul 17 '17 at 20:24
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Louis can do the following:

1) Align the ruler along one of the longer sides of the box (one end at a vertex of the box) and add a mark on the ruler at the far vertex.

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2) Align the ruler along a second (perpendicular) long side and confirm that it is the same length. At this stage Louis now knows the box is of dimensions AxAxB for some unknown A and B.

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3) a] Align the ruler along one length A side (as before). b] Now pivot the ruler 90 degrees about the end vertex so it is aligned with a length B side. c] Pivot 180 degrees about the new end vertex so the ruler is running back along a side of length B - the ruler will end short of the new far vertex, so d] now pivot the ruler 180 degrees about it's own end so it now extends past the box vertex. Repeat steps c] along side of length B three more times at which point the end of the ruler should be exactly at the far vertex.

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4) Carry out a similar process, this time running along a length A side 7 times (flipping the ruler as in step d] above twice, as necessary).

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If Jack is being honest about the box dimensions and the length of the ruler then Step 3 will show that A + 5*B = 2*35 and Step 4 will show that 7*A = 3*35. So from step 4, A = 3*35/7 = 15, and then from step 3, 15+5*B=2*35 so, B = (70-15)/5 = 11. If Louis didn't have a pencil to make the mark in step 1 then he could just pivot and rotate the ruler in a similar manner about the two long edges to confirm they were the same length, and reach the same conclusion.

Figure below shows approximately how he would do it.

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enter image description here

So Louis doesn't need the Coke can, the A4 paper, a Euro note, or even a pencil to confirm dimensions of the box.

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  • $\begingroup$ Wow. Very creative thinking Penguino $\endgroup$ – DEEM Jul 18 '17 at 12:33
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Here is my answer:

I suspected that both the 10 Euro bill and the A4 paper where useful as they have specific dimensions. A4 paper: 297mm x 210mm. 10 Euro bill: 127mm x 67mm. Assuming that there are at least a few piece of A4 paper, then the following is possible.

Arrange two piece of paper over the wooden ruler so that one piece is vertical and the other piece is horizontal. These combined lengths give us a length of 297mm+210mm = 507mm. Lets mark the wooden ruler here. We know that the ruler is 35 inches long which is equal to 889mm. So now we have a ruler that can measure 507mm = 19.96 inches on the one side and 889mm - 507mm = 382mm = 15.04 inches on the other. Now we can measure the sides that are 15 inches within the error margin.enter image description here[[1]

To be able to measure the 11 inch side we need to use both the A4 paper and the 10 Euro bill. First we place two horizontal pieces of paper starting from the line we drew earlier on the wooden ruler, towards the side the is 507mm. The we place 3 vertical pieces of paper starting from the end of the horizontal papers that we places and towards the line we drew. We will then see the the last vertical piece of paper will be slightly over the line we drew and this distance will be equal to (3*210) - (2*297) = 630-594 = 36mm. If we then place the 10 Euro bill vertically next to the vertical papers then we get the extra length of 67mm. This together with the line drawn and the overhang from the papers give us a distance of 507+36+67 = 610mm. If we remove this length from the total length of the wooden ruler then we will get 889mm - 610mm = 279mm = 10.98 inch. Now we can measure the side of the box that is 11 inches inside of the error margin. enter image description here

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So we have:

  • iPhone 6 plus: 6.22 * 3.06 * 0.28(0.302), screen 5.5
  • A4 paper: 8.27 * 11.7
  • ruler: 35
  • 10 euro note: 5 * 2.6378
  • coca cola can(US size taken from Wikipedia): 4.83 * 2.13 * 2.60

And that's how Louis proved it:

- measure the diagonal of the square, mark it on a ruler(21.21 normally, Pythagoras theorem: $sqrt(15^2 + 15^2) = 21.21$, other part will then be 13.79)
- measure other diagonal, must be same(if not same - it's not a square)
- put the ruler to each side that is supposed to be 15 and check that distance from the end of the side till the marking(at 21.21) is 15(14.99), by putting his iPhone 6 Plus(6.22 inches high, $21.21 - 6.22 = 14.99$) next to the box
- then put his iPhone and a can to the side that is 11, to measure that it is 11+-0.05 inches($6.22 + 4.83 = 11.05$)

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