# Who joined when?

An institute runs batches training students in C#. A new batch starts on the 1st of every month and every batch lasts for exactly 3 months (thus at any given time, the students of exactly 3 batches will be present concurrently). The table below gives the information for the total number of students studying at the institute in each month of 2018 (some values have intentionally been left blank).

It is also known that 95 new students joined in May, while 91 new students joined in September.

Find out how many joined when and how many left when ? Please share your approach.

you can use the structure like Month|total students| New joinee |leaving student

Source : IMS https://www.imsindia.com/

• Where does this puzzle come from? – Gareth McCaughan Oct 19 '18 at 15:12
• "IMS" could stand for almost anything, and nothing I see in my first page of search results makes it obvious what it is. Could you please be a lot more specific? If the thing is on the internet, make a link; if it's from a printed thing, say what. Thanks! – Gareth McCaughan Oct 19 '18 at 15:17
• Thanks for the link to imsindia.com -- but this is still extremely vague. Is this from some sort of ongoing competition? Are you copying things from their printed materials, or from their website, or what? ... And do you have their permission to reproduce their questions here? – Gareth McCaughan Oct 19 '18 at 15:24
• In what possible sense are you not copying their material? It's right there in what you posted. Of course you're copying their material. I'm asking whether you have their permission to do it. – Gareth McCaughan Oct 19 '18 at 15:47
• If there's no single solution and no definitive answer for some months like @gabbo1092 and Weather Vane noted in their answers/comments. What answer are you looking for? – JGibbers Oct 19 '18 at 18:51

This is the starting point with g and k known.

    Old  Cur  New
Jan  a +  b +  c = 283Feb  b +  c +  d = 284Mar  c +  d +  e = 286Apr  d +  e +  f = 284May  e +  f + 95 = 282Jun  f + 95 +  h = 284Jul 95 +  h +  i =  ?Aug  h +  i +  j = 283Sep  i +  j + 91 = 276Oct  j + 91 +  l = 273Nov 91 +  l +  m = 278Dec  l +  m +  n = 279

By subtracting Sep from Aug we know h = 98 so now we have
 Jun  f + 95 + 98 = 284
so we know know f = 91 and this can be now worked back to Jan to give
    Old  Cur  New
Jan 96 + 94 + 93 = 283Feb 94 + 93 + 97 = 284Mar 93 + 97 + 96 = 286Apr 97 + 96 + 91 = 284May 96 + 91 + 95 = 282Jun 91 + 95 + 98 = 284Jul 95 + 98 +  i =  ?Aug 98 +  i +  j = 283Sep  i +  j + 91 = 276Oct  j + 91 +  l = 273Nov 91 +  l +  m = 278Dec  l +  m +  n = 279

One more can be found, by subtracting Nov from Dec we know n = 92
Dec  l +  m + 92 = 279

But as far as I can see the rest can only be quess-work.

• I think after this point, you can give an upper and lower limit on i (185 and 3 respectively, if my numbers are correct), and each of those lead to a different solution. – elias Oct 19 '18 at 19:12
• It is correct Weather Vane. Thanks for your effort. – sam Oct 20 '18 at 9:01

I believe this is a solution:

  |Total|New|Leaving
J| 283 | 93| 96  F| 284 | 97| 94  M| 286 | 96| 93  A| 284 | 91| 97  M| 282 | 95| 96  J| 284 | 98| 91  J| 289 | 96| 95  A| 283 | 89| 98  S| 276 | 91| 96  O| 273 | 93| 89  N| 278 | 94| 91  D| 279 | 92| 93


General approach:

I started by filling in the totals, number of students entered in May and September which lead to number of students leaving in July and November. I then started subbing in possible rough numbers based on the numbers known. Then I basically adjusted the numbers as necessary starting near the bottom where more was strictly defined and working my way up since Jan and Feb have more flexibility in adjusting without effecting the other numbers. After a few iterations eventually got to numbers that worked under all the restraints.

• Your answer looks correct. Can you pls share your approach? Especially for Jan and April. – sam Oct 19 '18 at 15:27
• Sorry, But your answer is wrong. For some months, we cannot find the no of new joinees or left students. – sam Oct 19 '18 at 15:46
• I'm sorry I don't quite understand what you are saying is wrong with the answer. Are you saying the problem is defined in such a way that you cannot find one definitive solution? – gabbo1092 Oct 19 '18 at 15:48
• Yes, for some months we cannot find definite soln – sam Oct 19 '18 at 15:49