# IMAT 2022 Question 10

There was this question in the IMAT logical reasoning section the previous year, and I just do not understand the logic behind the answer. Here is the question:

On a volcanic island, scientists plan to install sensors to monitor ground motions and predict volcanic activity.

The outline of the island is shown below. The shaded area in the middle is the volcano.

A sensor cannot be installed on the volcano itself but can be installed on any one of the other squares. Each sensor takes measurements from the square on which it is installed and the surrounding squares. See an example below.

area covered by sensor
area not covered by sensor
S-sensor installed
What is the minimum number of sensors that need to be installed to take measurements of the entire area of the island (excluding the volcano)?
A 20
B 10
C 16
D 6
E 22

I think I am able to cover the Island with 19 sensors. But the right answer is 20, what am I doing wrong here?

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I did it and got 20, did the same thing, but the 4 singular squares on the left and right I filled with one each assuming that you cant put a sensor on a place already covered by the sensor (it didnt say this though so?) i am a bit confused about the Number 17 youve put down - thats not needed
Could anyone else clarify how to solve this?

yeap you are right, 17 is not needed it is possible to cover with 18 then

yeap you are right, 17 is not needed it is possible to cover with 18 then

Hey this is how I did

I think you can’t put that 9 right next to the 7 or the 14 next to the 9 because for instance the area you wrote the 9 is already covered by the 7 if that makes sense.

Hi, thanks for your solution, I think it’s the one the author had in mind

`I think you can’t put that 9 right next to the 7 or the 14 next to the 9 because for instance the area you wrote the 9 is already covered by the 7`

I do not think it’s true, it’s impossible to cover all the squares and not overlap sensors. For example, in your solution sensors around the black square are not actually lines and they do overlap with T-shaped sensors if you would draw them in full version.

Oh wow yes you’re right. It can be covered with 18 sensors although it’s not in the options. Such a weird question.

I think the logic was that you cant place the actual sensor in an area already covered (but the areas covered by sensors can overlap) so the sensors cant be directly next to each other? Thats the only way this question works with 20, feel like they should’ve said this in the question because you’re absolutely correct 18 would be the answer.

you cant place the actual sensor in an area already covered (but the areas covered by sensors can overlap)

but if two areas are overlapping, it means that at some point you put a sensor in an area that was already covered, in essence, there is no difference

At this point, I would cautiously assume that the authors made a mistake in this question.

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I did this one mathematically, but I’m not sure it’s correct. To me, it makes sense to look for a more general solution instead of trying to figure out specific cases (like looking for each combination of positions for the sensors), so I’ll give it a go here and tell me what you guys think, please!!

The whole area has 81 squares. Minus the volcano area, we have 72 squares to be covered. Each sensor can cover only 5 squares, which gives us the amount of 72/5 = 14.4 sensors. But the number of sensors must be an integer! So this 0.4 extra for each of the 14 sensors gives us 0.4*14 = extra 5.6 sensors, which in total represent 14.4 + 5.6 = 20 sensors

I think it’s a promising way to tackle this problem overall, I started the same way But there is a logical error in your reasoning. 14.4 sensors means that it is certainly not possible to cover the area with less than 15 sensors.

Meanwhile, 15 sensors could theoretically cover 15*5 = 75 squares, which is more than we have.

So we inevitably come to the question of exactly how to put the sensors. We have already proven earlier that it is possible to do so with 18. So the answer is somewhere between 15 and 18.

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Oooooh, yeah, you’re right! My ‘mathematical’ approach has a logical flaw. BUT now that I’ve read the other replies in more detail, I think what @SohamPaikaray said is actually correct. You said,

and that is NOT true! All sensors in this scheme (same as this guy’s: https://www.youtube.com/watch?v=r-djUVpqb_U) are put in blank squares, only their covered areas are the ones that overlap.

So yeah, I also think they should have written that it was mandatory for two sensors to not be placed directly near each other, because that’s the only way the answer is 20. Otherwise, 18 should also be a valid answer.

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Ah ok, I see what both of you meant. Yeah, you are right, it is working only with this limitation. Thanks for the discussion!