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I am trying to solve a mathematical arrangement based puzzle and not able to do it, need help of you guys. Question is relevant to competitive exams in India.

Here is the puzzle:-

M, N, O, P, Q, R and S are seven people who live on seven different floors of a building but not necessarily in the same order. The lower most floor of the building is numbered 1, the one above that is numbered 2 and so on till the topmost floor is numbered 7. Each one of them have different income i.e. 3500, 15000, 7500, 9000, 11000, 13500 and 5000. (But not necessarily in the same order.)

  1. M lives on an odd numbered floor but not on the floor numbered 3.
  2. The one who has income of 11000 lives immediately above M.
  3. Only two people live between M and the one who has income of 7500.
  4. The one who has income of 15000 lives on one of the odd numbered floors above P.
  5. Only three people live between O and the one who has income of 15000.
  6. The one who has income of 7500 lives immediately above O.
  7. R earns 4000 more than Q.
  8. The one who has income of 3500 lives immediately above the one who has income of 5000.
  9. Only one person lives between N and Q.
  10. N lives on one of the floors above Q.
  11. Neither O nor M has income of 9000.
  12. Q does not have income of 7500.

Please help me solve this arrangement and if explained step by step of your approach will be a great help.

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  • $\begingroup$ If your question is part of an ongoing contest, then we can't solve it for you. Sorry, but site policy is against allowing people to use us for cheating :-) Please clarify whether this is a competitive exam that you're currently taking or just a past paper. $\endgroup$ – Rand al'Thor Aug 23 '17 at 9:56
  • $\begingroup$ this an example question and not taken from any ongoing competitive test/exam.. $\endgroup$ – Sanny Srivastava Aug 23 '17 at 9:59
  • $\begingroup$ Ah, OK. I'll take a look at it then :-) $\endgroup$ – Rand al'Thor Aug 23 '17 at 10:03
  • $\begingroup$ thank you, also for formatting and tagging. $\endgroup$ – Sanny Srivastava Aug 23 '17 at 10:06
  • $\begingroup$ Could you double-check clues 7 and 12? The latter seems redundant, since if R earns 4000 more than Q, then Q can't earn 7500 anyway (as nobody earns 11500). $\endgroup$ – Rand al'Thor Aug 23 '17 at 10:40
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Logic grid

When faced with a problem like this, you can start by making a logic grid, plotting each pair of the attributes (letter, floor, income) against each other and crossing out the excluded possibilites. Here's our initial logic grid for this problem, reasoning directly from each clue in turn:

initial grid

Step-by-step solution

By clue 1, M must live on floor 1, 5, or 7. By clue 2, M can't be on the top floor, so M is on 1 or 5.

If M is on floor 1, then by clue 3, the 7500-earner must be on floor 4. By clue 6, O is on floor 3. By clue 5, the 15000-earner is on floor 7. Also, by clue 2, the 11000-earner is on floor 2. Now look at clue 8: we've already filled in enough floors and incomes that the only possibility for 3500 and 5000 being adjacent is that the 3500-earner is on floor 6 and the 5000-earner on floor 5. The only incomes left are 13500 and 9000, which must be on floors 1 and 3, i.e. they must be M and O in some order. By clue 11, contradiction!

So M is on floor 5. By clue 3, the 7500-earner must be on floor 2. By clue 6, O is on floor 1. By clue 5, the 15000-earner is on floor 5. Also, by clue 2, the 11000-earner is on floor 6. Now look at clue 8: we've already filled in enough floors and incomes that the only possibility for 3500 and 5000 being adjacent is that the 3500-earner is on floor 4 and the 5000-earner on floor 3. The only incomes left are 13500 and 9000; by clue 11, O is the 13500-earner on floor 1 and the 9000-earner is on floor 7.

Let's check our progress:

soon finished

Now consider N and Q. By clues 9 and 10, N is exactly two floors above Q. By what we've filled in so far, this means they must be either on floors 4 and 2 or on floors 6 and 4. But the 7500-earner is on floor 2, so by clue 12 (which is actually redundant given clue 7), Q can't be there. So N is on floor 6 and Q on floor 4. By clue 7, R must be the 7500-earner on floor 2.

Now we only have floors 3 and 7 to fill in. By clue 4 (since the 15000-earner is on floor 5), P must be on floor 3 and S on floor 7. Finished!

Final answer

M earns 15000 and is on floor 5.
N earns 11000 and is on floor 6.
O earns 13500 and is on floor 1.
P earns 5000 and is on floor 3.
Q earns 3500 and is on floor 4.
R earns 7500 and is on floor 2.
S earns 9000 and is on floor 7.

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  • $\begingroup$ Quick answer with great explanation, thank you a lot. I am trying to understand the GRID structure you have created as never seen such before and it is looking very effective. Just a quick question, is it worth creating this huge grid in an exam where time is the key? $\endgroup$ – Sanny Srivastava Aug 23 '17 at 11:34
  • $\begingroup$ @Sanny That depends on how well you can make these deductions in your head. I don't normally bother making a grid for puzzles like this, because I can find a weak spot and start deducing until it all falls apart (general advice for puzzles like this: find something about which we're given a lot of information - e.g. here, M appears a lot in clues, so start by making deductions about M - because lots of information enables us to narrow down the possibilities a lot), but if you've had less practice or can't keep all the clues in your head at once, then it might be worth drawing a grid. $\endgroup$ – Rand al'Thor Aug 23 '17 at 11:40
  • $\begingroup$ thank you again, you are great in answering questions. It would have been great if I could invite you to answer all my future questions. :) :) $\endgroup$ – Sanny Srivastava Aug 23 '17 at 11:46

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