# Building towers with colored blocks

You have blocks in five colors (red, green, yellow, blue and black - or 1 to 5). Each color comes in two varieties: a cube and one as big as two cubes stacked atop each other (sizes 1 and 2).

You are supposed to use them to build two towers next to each other such that two blocks of the same color (in either tower) are separated by the sum of their heights. For example:

Note that this only says that each color must repeat within a given distance, but not that every block of the tower must be of a particular color, so you get white blocks to fill any holes (they don't do anything else, and they aren't subject to any of the rules above or below).

You don't know how high the tower is supposed to be, hence give the shortest repeating pattern that can be used to extend such a tower upwards for any height. Within that pattern, each block of each size and color should repeat an equal number of times (ignoring white, of course).

Now if that were all, finding a solution would be simple, e.g. this one:

However, in the pattern of each tower each block size of a color should be opposite a block of each other color in the other tower. The above example doesn't fulfill this: consider the blue 2-block: while it occurs on the left opposite to the black 2-block, but never vice versa, and never opposite to the green 2-block, nor does the blue 1-block occur opposite the black 1-block.

Thus for any pair of two colors (not involving white and not both colors the same) the following patterns must occur (given as example for red and blue):

Note that in 3 and 4 they don't need to be perfectly lined up and in 5-8 the small one doesn't need to be at the same height, but there must be one box in which they are opposite, so e.g. for 3 and 5 these are alternative examples:

• Those Ascii pictures are really hard to understand, could you please replace them with MS Paint (or something like it) pictures? Apr 25, 2015 at 9:33
• When checking validity, it looks like we are supposed to alternate towers looking at the same colors, and the distance between the blocks of the same color must equal the distance of the heights of the blocks. This property must hold for all blocks (except white) in both towers, right? But, from the 1st examples it seems like all blocks of the same color in the same tower must also meet the distance requirement, but then, in the displayed solution, the distance between the red blocks in either of the towers is not equal to their sum, so I'm confused.
– JLee
Apr 25, 2015 at 13:49
• This problem looks very very hard, I like it! Apr 25, 2015 at 14:12
• feeling some ambiguity in reconciling these: "each block size of a color should be opposite a block of each other color in the other tower" and in the comment above: "it doesn't always have to be opposite a block of the same size, only sometimes"
– JLee
Apr 26, 2015 at 21:39
• Are there any restrictions on the number of blocks we can use? That is, do we need an equal number of 1x1 and 2x1 blocks of each color? May 28, 2015 at 19:55

## An Example with Two Colors

This is impossible.

I can find a pattern for matching little blocks to little blocks, big to big, and big to small. The (small x small) and (big x big) towers fit together. However, it is not possible to connect the mixed tower. This stems from the fact that adding a big block adds +1 spacing but also adds +1 spacing requirement so the net gain is 0. I feel like I'm playing chess and I'm trying to get a bishop to the opposite color square.

## An Example with Three Colors

I believe this may not be possible.

Again, I can find small x small and big x big that all connect but the mixed, although it is valid by itself, can't be connected to the rest.

## An Example with Four Colors

I believe this is also impossible.

Again, I can find small x small and big x big that all connect but the mixed can't be connected to the rest. In fact, I couldn't even finish the mixed tower.

I was unable to integrate the following pairs into the mixed tower:

## Weak Lower Bound for Five Colors

If I understand the problem correctly, then a weak lower bound would be 60 blocks tall.

What's shown below complies with the rule about pairs of colors being opposite each other but with no regard to the rule about them being spaced apart by the sum of their heights. There are no repeated pairs and no white spaces so I am confident that this is the lower bound. If it is possible to rearrange these to meet the final rule, then the solution would be complete. Otherwise, we must begin inserting white spaces as needed. (The tower is shown broken into pieces that can be stacked upon one another without any white space. This is merely for formatting.)

However, even the four-color pattern is 131 blocks tall and it doesn't even meet all the requirements. I suspect that the five-color solution (if it exists) will be taller.

• nevermind, i now see that you are saying that your answer is not even close to complete, right?
– JLee
Jun 24, 2015 at 21:36
• @JLee good catch on the typo. I just wanted to bump the question and show the pattern has to be at least 60 blocks tall. Jun 24, 2015 at 21:36
• @JLee Right. This is just a possible first step. Jun 24, 2015 at 21:37
• I hope you're right, but I don't even see that it is that! This is the hardest puzzle that I have ever seen. The search space is so huge that I cannot imagine that a computer would be of much help. So, I'd love to see it solved.
– JLee
Jun 24, 2015 at 21:38
• Good luck, and may supernatural forces be with you
– JLee
Jun 24, 2015 at 23:43

Let's assign each block a color field.

A block's color field encompasses both towers and spans three block lengths (for both singletons and doubles) - the block itself and one span beyond either end.
The rules of tower-building are then equivalent to the following:
- Every segment of both towers must be covered by color fields of every color.
- Color fields of the same color may not overlap.

From this it is plain to see that each color must cover the tower with density $$\frac{1}{6}$$.
Moreover, since sections of color field are either 3 or 6 segments long, the blocks of any individual color are constrained modulo 3. In fact, all singletons of a color have the same height modulo 3, and doubles of that color must occupy the other two residue classes.

This means that singletons of two colors can only match each other if their color fields match modulo 3, and singletons and doubles can only match each other if their color fields do not match modulo 3. Therefore, it is impossible for every color to meet every other color in all possible ways.

• Is this saying that singletons are possible and that doubles are possible, but that both together are not possible?
– JLee
Jun 10, 2022 at 9:29
• For any two colors A and B, A's singletons can either match B's singletons or B's doubles, but not both. Jun 10, 2022 at 13:51