# How to find a volume of this figure (which is $3080 \text{ cm}^3$) in a few seconds?

I was watching this Japanese game show and came across this question:

The contestants were told that each small cube is 2cm on its side and were asked to find the volume of the above figure.

The answer was 3080 $cm^3$.

While I was counting the number of cubes for the first row, one of the contestants was able to answer this within a few seconds.

I'm curious about how he did it. I assumed the figure was constructed in some sort of pattern and was hoping someone could shed some light on this.

(The game show didn't explain how to solve this unfortunately...)

• The back left face has $77$ cubes in it. By looking at the dark squares you can see that the layer next to it has $7$ cubes fewer, that is $70$; and the next has $9$ cubes fewer, that is $61$; and so on. We get$$77+70+61+\cdots=385$$cubes with volume $385\times8=3090$. But I really don't think I could do this in a few seconds. Obviously Japanese game show contestants are smarter than me... – David Feb 16 '18 at 5:57
• @David - seeing the YouTube video and the suggested related videos, it seems that one particular Japanese game show contestant is fast and has an incredible memory - the others do not seem to get a look in – Henry Feb 16 '18 at 9:30
• Is there a chance that in fact the host did declare that it is a 'even pyramid' and thus is some certain fraction (which I don't know!) of simply the overall cubic shape ?? – Fattie Feb 16 '18 at 19:22

I looked at the horizontal layers.

Top layer has seven, and each layer below shows seven more. So the number of cubes is $$7+14+\cdots+70=\frac{77}2\cdot10=385\,.$$

• I couldn't even count to seven in the time it took the contestant to answer, let alone count each row and notice a pattern. – Jack M Feb 16 '18 at 16:27
• Yes, @JackM, and it didn’t help me that I first miscounted the number in the top layer. – Lubin Feb 16 '18 at 17:43
• Maybe that contestant is just exceptionally good at recognizing numbers visually, and is able to immediately visually recognize a group of seven objects in the same way that most people can visually recognize two or three. – Jack M Feb 17 '18 at 7:57
• I think the ability to recognize numbers is exactly the key to that person’s speed, @JackM . – Lubin Feb 17 '18 at 18:20
• If they had shaded the top faces of the cubes instead of the right faces I might even have got that - see my comment above :( – David Feb 18 '18 at 23:32

Here's my idea for how someone could answer this in a few seconds.

1. See that there are ten horizontal layers
2. See that each layer adds seven blocks
3. Know that the tenth triangular number is $55$.
4. Know that $2 ^ 3 = 8$
5. Multiply $8 \cdot 55 \cdot 7$
• You'd need step 3 to know step 2 is relevant. – Pete Kirkham Feb 16 '18 at 11:43
• @PeteKirkham You're right. I put it as step two so it'd be obvious where "tenth" comes from. I'll change the order--hopefully it will still be apparent that step three relies on both previous steps. – GoalBased Feb 19 '18 at 2:14

So we start on the left, and kind of slice it diagonally, if it makes sense.

The first diagonal layer has TWO columns, one with $2$ blocks and another with $2$ blocks.

The second diagonal layer has THREE columns, one with $4$ blocks, another with $3$ blocks, and another with $3$ blocks.

The third diagonal layer has FIVE columns, with $6$, $4$, $2$, $2$ and $1$ blocks.

If we sum it up to the tenth diagonal layer, we end up with a total of $385$ blocks.

EDIT: Didn't see the pattern.

• Seems unlikely someone could do that in a few seconds? – IntegrateThis Feb 16 '18 at 5:56
• people are crazy, this woman gave the 23rd root of a 201 digit number in 50 seconds – Saketh Malyala Feb 16 '18 at 5:57
• i don't even think i can write 200 numbers in 50 seconds bro – Saketh Malyala Feb 16 '18 at 5:58
• fair enough lol. – IntegrateThis Feb 16 '18 at 5:58
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