The challenges of clear communication: Not everyone thinks like you

How many blocks are needed to complete a full cube?

This image pops up from time to time on social media. I posted this puzzle on my Facebook feed and I got a whole load of different answers and questions that were really surprising and generated quite a bit of debate.

This simple activity got me thinking about the way we think about things and the challenges of clear communication. 

First, have a look at the question and picture. What’s your answer?

The “official” answer is 45. A cube has equal lengths on each side. So to reach the answer of 45, you need to fill in the slope: 3×4 + 2×3 + 1×2 = 20, and then add on the top layer to finish the cube as it is only 4 blocks high, so that’s another 5×5=25. So 20+25 = 45. 

Good. So how did other people respond?

“What is a full cube?” Yes, the question could be written better, and before you even get to counting blocks, you are wondering if it is “this” cube or “any” cube.

Assuming “this” cube, another question I got was “what does complete mean?” This was quite interesting. Some people jumped right in and started counting blocks, but others were stumped. So there is an implicit assumption in the problem design that people will understand the mathematical language.

Once I explained that you needed to work out how many extra blocks you would add to build a cube , those people got on with solving the problem.

How did people get on with solving it? Well, a lot of people did not spot that the drawing is only 4 blocks high. So they missed out that they needed a top layer, and were short by 25 blocks. 

It’s a great drawing for giving you the illusion and confidence that you know what you are looking at, leading the eye to focus on the long sides at the top and front. 

Things then took a philosophical turn. “What’s around the back of the block? You can’t solve this puzzle without knowing that.” I was really surprised about how common this question was. Maybe an oddity of my Facebook friend group. But it surprised me because when I had looked at the picture, I had assumed it was solid and the blocks were stacked up. Therefore, I reasoned, it had to be completely filled to be structurally sound. 

But what if the blocks were glued together? Well, then there could be any number missing around the back and you would certainly need more than 45. There are around 40 visible blocks, so to complete a 5x5x5=125 block cube, you would need around 85 blocks if it was completely empty around the back.

“Infinity.” This was surprising, but also revealing. Most people assumed they needed to make a 5x5x5 block to be complete. But in reality, the cube could have sides of any length! Go large! Even to infinity!

“Zero.” Huh? Well, the person explained, I’ve already got lots of blocks, so I will take away blocks and make a 4x4x4 cube or a 3x3x3 cube or a 2x2x2 cube or a 1x1x1 cube, and then I have a complete cube! I salute this independent thinking!

So, if you really want to write this puzzle question correctly, it has to be “How many blocks will you add to build a cube of 5 x 5 x 5 blocks?”

And the correct answer? 0, 45, or any cubed number with a side of length 6 or more.