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Yesterday afternoon the sun was shining into my room and making some shadows of different objects on my table. By looking at the shadows I decided to make a puzzle for this community. The picture below is what I arranged on my table and I want you to find the height of the pencil. enter image description here

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  • $\begingroup$ Can we assume that similar objects are of equal size? $\endgroup$ – feelinferrety Oct 17 '17 at 23:36
  • $\begingroup$ @feelinferrety, Yes of course they have equal measurements. $\endgroup$ – Seyed Oct 17 '17 at 23:38
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    $\begingroup$ We can't always trust our eyes, especially in the world of puzzles. We also can't always trust our assumptions, especially in the world of mathematics. It is best to include explicit statements for any such points. $\endgroup$ – feelinferrety Oct 17 '17 at 23:43
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The height of the pencil is

14cm.

To get this, I found what the length of the shadow would be if the wall were not there:

Sketch of problem

From the pencil sharpener, we know that the ratio between object height and shadow length is

2:3, that is, a 2cm object would create a 3cm shadow.

Using given lengths, the total height of the shadow is

h' = 8cm (3cm box and 5cm above the box)

That means that if the wall weren't there, the shadow would go

L' = 12cm past the wall (Note the similar triangles in my sketch above)

Using the lengths given, and the fact that the matchbox must be the same depth as the pencil sharpener (they produce the same length shadow), we can find what the total length of the shadow would be without the wall

L = 12cm + 8cm + 1cm = 21cm

Now, we just use the original ratio between shadow length and object height to find a final height of

h = 14cm

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  • $\begingroup$ Excellent work :-) $\endgroup$ – Seyed Oct 18 '17 at 0:42
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answer...

14cm

this is because the shadow has two parts, horizontal would be proportional, but vertical would be similar. By that I mean, the vertical portion of the shadow is parallel to the pencil, hence it must be generated from a length of pencil equal to itself.

vertical portion = 5cm + 3cm = 8cm (A)

Now for the horizontal portion, notice the width of the sharpner. The shadow is in the ratio of 1.5:1 w.r.t. the body.

Hence the horizontal portion of the pencil's shadow must also be in the same ratio. Since the horizontal portion of shadow is 8cm + 1cm = 9cm.

pencil portion casting horizontal shadow is 9/1.5 = 6cm (B)

Hence total length of pencil is:

(A) + (B) = 8 + 6 = 14 cm

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  • $\begingroup$ This has the same final answer as, and essentially equivalent reasoning to, DqwertyC's answer posted 13 hours before yours (and I think also accepted then by the questioner). Before posting an answer, please look at other existing answers and check whether yours adds something new. (If you think it does and might not be obvious, it's worth being explicit about what and why.) $\endgroup$ – Gareth McCaughan Oct 18 '17 at 15:19
  • $\begingroup$ DqwertyC had taken an approach of extrapolating the shadow and laying it all flat on horizontal surface. On the contrary, I split the two shadows portions, vertical and horizontal, and came to the conclusion.. However the underlying assumption of shadow ratio being 1:1.5 would inevitably be the same... $\endgroup$ – murphy1310 Oct 19 '17 at 17:49

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