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An Egyptian pharaoh wants to build a monument to the Sun God, Ra. He wants this to be a solid stone cube, which is $20$ hectocubits tall, long and wide.

His engineers get right to work planning this. The only stones the Egyptians have access to are $2000$ enormous rectangular blocks leftover from a failed project, whose dimensions are $2\times 2\times1 $ (all units in hectocubits), and which cannot be broken into smaller usable blocks. They figure that the easiest way to build this monument is in horizontal layers, each layer being a 10 by 10 array of blocks which lie flat. They present this plan to the pharaoh.

However, after seeing the plan, the pharaoh is furious! Since the edges of the blocks are lightly rounded, the cracks between blocks can let some light through. This is of course a problem, since the cube is supposed to be an homage to the generosity of the Sun God, so it should graciously accept the gift of the Sun's rays instead of letting them pass by.

The engineers get confused at this point, since they aren't as religious, and don't understand the importance of the pharaoh's request. Totally losing his cool, the pharaoh yells, "You fools! All I ask is that you

build a $20\times20\times20$ cube out of $2\times2\times 1$ blocks so no light can shine in one face though the cracks between the blocks and exit the opposite face!

IS THAT SO DIFFICULT!?"

Can the engineers succeed?

Note that the pharaoh wants the cube to block all possible beams of light, not just vertical ones. In other words, every line which enters one face and exits the opposite face must also pass through the interior of one of the $2\times2\times1$ blocks. The solution does not involve lateral-thinking, so the answer is not "just fill the cracks with mortar."

Edit: I changed it so that the cube is only required to block light which enters and leaves opposite faces. As @frodoskywalker pointed out, it was trivially impossible without this addition. Furthermore, I clarified other confusing points (breaking blocks is not allowed, and the cube must be solid, implying all the blocks are axis aligned).

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  • $\begingroup$ Am I right in assuming this turns the question into 'can you fill a 20x20x20 cube with 2x2x1 blocks so there is no single line across the cube that lies entirely along block edges'? The 'not perfectly rectangular' part is a big vague to interpret otherwise. $\endgroup$ – Tim Couwelier Jun 3 '15 at 7:57
  • $\begingroup$ @TimCouwelier Your interpretation is correct. $\endgroup$ – Mike Earnest Jun 3 '15 at 8:20
  • $\begingroup$ Is Pharaoh concerned about rays from the midday sun reaching the ground through cracks between the blocks? $\endgroup$ – Bob Jun 3 '15 at 8:27
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    $\begingroup$ Is your requirement that "every line which pierces the interior of the cube must also pass through one of the smaller blocks" precisely what you meant? Or do the lines need to enter and exit through opposite faces, have a minimum length within the cube, etc...? $\endgroup$ – frodoskywalker Jun 3 '15 at 11:12
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    $\begingroup$ Do the interior blocks all have to be axis-aligned? $\endgroup$ – Random832 Jun 3 '15 at 13:27
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The engineers can't do it, and here's why:

Let's consider an arbitrary arrangement of blocks. Consider the $19\cdot 19\cdot 3=1083$ axis-aligned lines (let's call those important) with integer cubit offsets passing through the monument. Every block in the monument can block at most one such line. (1)

Claim: It is impossible for any important line to pass through exactly one block.

Proof of claim: Assume there's an important line passing through exactly one block $B$. Let's consider the two axis-aligned planes containing this important line. These planes dissect the monument into four cuboids.

Now let's look at the volume of one of these cuboids, call it $C$. Since one of the side lengths of $C$ is 20 cubits and the other side lengths are integer amounts of cubits, the total volume must be an even number of cubed cubits.

On the other hand, $C$ is composed out of parts of blocks. Since $B$ is the only block intersecting both planes, it contributes one cubed cubit, but every other block contributes either zero, two or four cubed cubits to $C$. Hence, the volume of $C$ is an odd number of cubed cubits.

Now we have our contradiction, hence the assumption is false and the claim is proved.

Now to block light going through all the important lines we need at least one block blocking each important line, and by the claim, we need at least two blocks blocking each important line. By (1), all these blocks are distinct, so we need 2166 blocks in total, but since we have only 2000 blocks, there will always be at least 83 unblocked important lines, no matter how we build the monument. $\square$

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  • $\begingroup$ Nicely done! Welcome to puzzling.stackexchange.com $\endgroup$ – frodoskywalker Jun 3 '15 at 12:32
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    $\begingroup$ Very nice! Two questions; Is 83 unblocked lines achievable? Also; What about arrangements that does not fill the cube? (The last question is in light of Random832's comment to OP) $\endgroup$ – Taemyr Jun 3 '15 at 14:09
  • $\begingroup$ This is a very clean proof, well done! $\endgroup$ – Mike Earnest Jun 3 '15 at 15:42
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The best I could do is this:

http://i.imgur.com/ggNPDre.png

You could just stack this x10 and it would use exactly the given 2000 blocks, but it does not fulfill the criteria so I am just going to leave this here in case someone gets an idea from this.

Actually now that I thought more about it just rotate it 90 degrees every other layer and that should work.

Even more thinking and Taemyrs comment proved that it won't work because of horizontal light

Some math

The squares sides are 10 9 10 9 each being 2 units tall
10x10x2 = 200 (2 identical sides)
It loses 1 block each time making it
9x10x2 = 180
8x10x2 = 160
...
1x10x2 = 20
20+40+60+80+100+120+140+160+180+200 = 1100

And the other sides start with 9 blocks so they have the exact same except -200 which = 900
900+1100 = 2000

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  • $\begingroup$ This will not work. Light will come through at the points on the diagonal. See Bob's answer. Light can also come through in the horizontal plane. $\endgroup$ – Taemyr Jun 3 '15 at 10:33
  • $\begingroup$ What did you use to produce the illustration? $\endgroup$ – frodoskywalker Jun 3 '15 at 10:39
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    $\begingroup$ @frodoskywalker Paint with grid enabled $\endgroup$ – Slyre Jun 3 '15 at 10:44
  • $\begingroup$ While this works for one of your layers, it doesn't work for the space between the layers. $\endgroup$ – Tim Couwelier Jun 3 '15 at 10:58
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    $\begingroup$ You've done exactly what the architects did to make the pharoah angry "They figure that the easiest way to build this monument is in horizontal layers, each layer being a 10 by 10 array of blocks which lie flat." $\endgroup$ – Pharap Jun 3 '15 at 14:30
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Start in a corner. Stand a pair of blocks on edge so that they 2 units high and meet perpendicularly to form the corner. Lay a block flat on the ground inside the square so that it is snug up against these blocks. Light coming in the bottom unit of the diagonal is blocked by this flat block. If you were trying to make a 4x4 block you could plop another flat block down on top of the first and another pair of blocks around the other two sides and be done. But you're not.

Extend each side wall with another pair of vertical blocks. The seams you've just made are blocked (in the bottom unit) by the flat block that is already installed. Add three more flat blocks to tile the floor, which is now 4 flat blocks. Return to the corner and put vertical blocks inside the side wall. These are one unit further "in" than the outermost wall, so will block the top halves of the seams. There is still a diagonal seam right on the corner in both the inner and outer vertical wall. Lay another flat block inside to cover it.

Repeat the pattern at the other corners and work back towards the centre until everything meets. You have two layers vertically on the outside of the whole cube. The vertical seams are blocked by flat tile inside. The horizontal seams in each vertical layer are blocked by the other vertical layer. After that it's all flat tiles. The ones in the bottom-most layer are offset 1 unit from the others to keep sunlight from getting through to the ground. At the top, do the same as you omit the final inner vertical course.

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