## Logicalympics — 100 meters!!!

Just as you go to the gym daily and increase your physical stamina, so also, you should go to the “mental gym” of solving hard math or logical puzzles daily to increase your mental stamina. You should start with a laser-like focus (or, concentrate like Shiva’s third eye, as is famous in Hindu mythology/scriptures!!) for 15-30 min daily and sustain that pace for a month at least. Give yourself a chance. Start with the following:

The logicalympics take place every year in a very quiet setting so that the competitors can concentrate on their events — not so much the events themselves, but the results. At the logicalympics every event ends in a tie so that no one goes home disappointed ðŸ™‚ There were five entries in the room, so they held five races in order that each competitor could win, and so that each competitor could also take his/her turn in 2nd, 3rd, 4th, and 5th place. The final results showed that each competitor had duly taken taken their turn in finishing in each of the five positions. Given the following information, what were the results of each of the five races?

The five competitors were A, B, C, D and E. C didn’t win the fourth race. In the first race A finished before C who in turn finished after B. A finished in a better position in the fourth race than in the second race. E didn’t win the second race. E finished two places Â behind C in the first race. D lost the fourth race. A finished ahead of B in the fourth race, but B finished before A and C in the third race. A had already finished before C in the second race who in turn finished after B again. B was not first in the first race and D was not last. D finished in a better position in the second race than in the first race and finished before B. A wasn’t second in the second race and also finished before B.

So, is your brain racing now to finish this puzzle?

Cheers,

Nalin Pithwa.

PS: Many of the puzzles on my blog(s) are from famous literature/books/sources, but I would not like to reveal them as I feel that students gain the most when they really try these questions on their own rather than quickly give up and ask for help or look up solutions. Students have finally to stand on their own feet! (I do not claim creating these questions or puzzles; I am only a math tutor and sometimes, a tutor on the web.) I feel that even a “wrong” attempt is a “partial” attempt; if u can see where your own reasoning has failed, that is also partial success!

### 33 Comments

1. Aditya Bhasin
Posted August 27, 2019 at 4:11 pm | Permalink | Reply

there is a question from logicalympics swimming competition similar to above I would like to discuss have solved myself without answer and checked after solving just like to discuss to further clear doubt.

2. Posted August 27, 2019 at 4:41 pm | Permalink | Reply

Hello Aditya Bhasin: please share your question and your detailed attempt/solution…I would love to discuss with you…regards Nalin Pithwa

3. Aditya Bhasin
Posted August 28, 2019 at 11:41 am | Permalink | Reply

four countries entered the logicalympic swimming events Great Britain (GB), the German democratic republic (GDR), USA and USSR. they were eight events in all 100m freestyle, 100 m backstroke, 100m breaststroke, 100m butterfly. 200m freestyle, 200m backstroke, 200m breaststoke and 200m butterfly. All four countries entered 3 swimmers in each of the 8 events. In a true logicalympic style there was no overall winner each of the four countries won 2 gold medals, 2 silver and 2 bronze. given following information see if you can work out which country won gold, silver and bronze in each event.
The USA didn’t win any 200m event and won no medals at all in any butterfly event. GB won one 100m and one 200m event. USSR was the only country to win all three medals in one event. GDR won only one medal out of both butterfly events, and did not win a gold and silver in the same event. GB won a total of two medals in breastroke events. GDR won a silver medal in one of the breaststroke events and won a total of two medals in the backstroke events. GB won a silver medal and a bronze medal in one of the 100m events. The USSR won a bronze medal in the 100m freestyle. GB won only one bronze medal of the 4 bronze medals in for the 200m events and did not win a silver medal in any of the 200m events. The USA won the bronze medal in both of the events where GB won the gold medal. the USSR won a silver medal in the 100m event won by GB which was not freestyle. the GDR won one 100m event and one 200m event. GBR won a 200m event in which the GDR won no medals at all. The USSR didn’t win a single medal in any of the backstroke events. the GDR won a bronze and silver in of the 100m events

4. Aditya Bhasin
Posted August 28, 2019 at 11:47 am | Permalink | Reply

A part missed out is USSR didn’t win any medals in the butterfly event

5. Aditya Bhasin
Posted August 28, 2019 at 1:36 pm | Permalink | Reply

The Answer is as follows:

100 M Freestyle USA (gold) GB (silver) USSR (Bronze)
100 M Backstroke USA (gold) GDR (silver) GDR (Bronze)
100 M Breaststroke GB (Gold) USSR (silver) USA ( Bronze)
100 M Butterfly GDR (Gold) GB (silver) GB (Bronze)
200 M Freestyle GDR (Gold) USA (silver) GDR (bronze)
200 M Backstroke GB (Gold) USA (silver) USA (bronze)
200 M Breaststroke USSR (Gold) GDR (silver) GB (bronze)
200 M Butterfly USSR (Gold, silver, bronze)

6. Aditya Bhasin
Posted August 28, 2019 at 1:51 pm | Permalink | Reply

My attempted solution was correct also but as follows just want to know if the reasons are the same for the medals.
To begin with USSR won the bronze medal in 100 M Freestyle as given by the question. Next USSR won a silver, in which, GB won the gold and USA would win the bronze as USA bronze is whenever GB has a gold. USSR won the silver in a 100 M event. But did not win any medals in Backstroke or Butterfly as per question. Also the event was not freestyle. Hence the 100 M event would be breaststroke which is Gold to GB, Silver USSR and Bronze USA (All info) in question.
Next GDR won a silver medal in one of the breaststroke events as it could not be the 100 M breaststroke it was the 200 M Breaststroke. Further info states GB won a total of 2 medals in breaststroke events (already one gold in 100 M. ) The medal GB won could not be gold as per question GB only won an event in which GDR had no medals at all hence it cannot be gold as GDR already has a silver in 200 M breaststroke. Hence it would be bronze ( and the only one of four bronzes) in 200m.
Next USA could not have won any 200M event and not any butterfly event as Breaststroke is already won by GB (in 100 M) USA wins the 100M freestyle and 100M Backstroke respectively. Gold for both. (No 200M, Not Butterfly or Breaststroke). GDR thus can only win the 100 M Butterfly in which they only won one medal. Am I right so far on the basis of Am I following the Q properly. Please provide some feedback.

Thanks and regards

Aditya

7. Posted August 28, 2019 at 1:58 pm | Permalink | Reply

I will certainly provide a detailed feedback to your own solution…Just give me some time to cook up my solution first…:-)

8. Aditya Bhasin
Posted August 30, 2019 at 10:50 am | Permalink | Reply

Hi Nalin, Have you done the answer yet. One small piece of info missing is USSR didn’t win medals in 100 M butterfly event

• Posted September 3, 2019 at 1:02 pm | Permalink | Reply

Sorry…I have to yet to work on it…To be frank, though this is not the appropriate place to share, I got entangled in a famous, popular major Hindu festival celebration last 3-4 days…it is an occasion to socialize, family-get-togethers etc…

• Posted September 3, 2019 at 1:04 pm | Permalink | Reply

Meanwhile, I was wondering…in fact, whole heartedly waiting for you to try other questions on the blog…and comment/feedback/…extensively…freely…I will certainly reply in detail to all further questions on other puzzles…regards

9. Posted August 30, 2019 at 1:01 pm | Permalink | Reply

Hello Aditya: I will have a look over the weekend. I will consider all the comments/observations you have made including the above one. Regards

10. Posted September 3, 2019 at 2:41 pm | Permalink | Reply

Hello Aditya Bhasin:

Let me first post the solution to the original puzzle I had posted: (Firstly, each of A, B, C, D and E is in the first place once and only once. So, also, each of A, B, C, D and E is in 2nd, 3rd, 4th and 5th places once and only once during the “whole competition of 5 races”. So, actually what we are seeking is a matrix of 5 by 5. (Now, it did occur to me that there might not be a unique solution to the question; anyways, I didn’t investigate this further). I started with encapsulating the given various conditions in to some form of inequalities (or perhaps, what I mean is symbols which I comfortable to play with): My ROOT assumption was let A be the first winner in the first event. Then, I think considering all the given conditions, I assumed let D be in first place of event 2. I continued this way; wherever it seemed to me that there were no conditions, or too many slots to be filled up, I made one assumption at a step (clearly noting it in my work). So here is the matrix that I got; it satisfies all the conditions; I request you to verify it for me and let me know) : Event 1: (first prize to fifth prize written left to right: ABCDE; Event 2: DEABC; Event 3: BCDEA; Event 4: EADCB; Event 5: CEBAD.

Well, please let me know if this is right.

11. Posted September 3, 2019 at 6:53 pm | Permalink | Reply

Hello Aditya: I have started working on a (possible) solution to the puzzle you posed. But, I also tried to incorporate your comments on Aug 28 and aug 30. On Aug 28, you have said that “a part missed out is USSR didn’t win any medals in the butterfly event; and on aug 30, you have said :”one small piece of info missing is USSR didn’t win medals in 100 m butterfly”. I also noticed that your solution says; USSR (200m butterfly) gets 2 gold, 2 silver, 2 bronze); does it mean that your in first correction on aug 28 itself your comment should have been more specific —- USSR didn’t win any medals in 100 m butterfly event. Else, it actually means USSR did not win in any medal in both 100 m and 200 m butterfly event also.

PLease clarify this once more. I will work on it afresh as soon as I receive your clarification. Regards, NP

By the way, your puzzle is much more interesting as it involves many more parameters/conditions and is a larger matrix. I like it. Thanks for sharing it.

12. Aditya Bhasin
Posted September 4, 2019 at 6:24 am | Permalink | Reply

Hi Nalin. Thanks for your reply. Yes My correction was that USSR did not win any medals in butterfly only in the 100 M event not all butterfly events. The Answer says that USSR won not 2 but 1 gold, silver, and bronze in butterfly 200 M event just as the question says they were the only team to win all three medals in one event. One more thing I would like to add is that when the question says that a team “won” an event it means they won the gold medal for e.g. “USSR won a silver medal in the event “won” by GB” means GB won the event winning the gold medal.Hope this information helps also.

13. Aditya Bhasin
Posted September 4, 2019 at 6:33 am | Permalink | Reply

I almost forgot the question iterates it entered 3 swimmers in each event hence only one gold, one silver, and one bronze. But overall in all of the events 2 gold, 2 silver and 2 bronze hence e.g. hence 2 gold in any of 2 different events e.g 100m gold and 200m gold etc. But in one event they is only one gold medal. one silver and one bronze but each team wins 2 gold, 2 silver and two bronze hence four teams 8 gold, 8 silver and 8 bronze all in different events. In one event even with 3 swimmers from each country there will only be one winner of gold, silver and bronze in each event.

14. Aditya Bhasin
Posted September 4, 2019 at 6:39 am | Permalink | Reply

…. and winning a medal is not the same as winning “the event” winning a medal could be either gold, silver, or bronze but winning the event actually means winning the gold medal. Thanks I hope this helps. I was also a little confused when beginning.

15. Posted September 4, 2019 at 7:22 am | Permalink | Reply

Oh…well, Aditya ..your clarifications arrive just in time…else I was trying to work out with whatever “additional assumptions I could make”…So, mostly, by tonight…my answer will be ready. Once again, I like the problem very much. Oh…by the way, it reminds that quite a few questions of this type appear in India’s CAT MBA entrance exams. Just FYI only sharing. Regards NP

16. Aditya Bhasin
Posted September 4, 2019 at 7:33 am | Permalink | Reply

Thanks Nalin, Look forward to your reply.

17. Aditya Bhasin
Posted September 4, 2019 at 1:27 pm | Permalink | Reply

Hi Nalin Sorry mind my lateness a very Happy Ganesh Chaturthi to you. I’m solving a few puzzles in other books and they are also tricky and time consuming I can share with you if you like but I’m not doing any extremely difficult mathematics. Yes the Q you posted (the 100 M) was also in the book I have and I had also attempted to solve it, it is all correct I will share in my next post how I had dealt with the question

• Posted September 4, 2019 at 5:08 pm | Permalink | Reply

Hello Aditya Bhasin: You are most welcome to share similar kind of (other) puzzles that you are reading from your book(s). We can compare our solutions/methodologies/approaches. I hope to solve that 8 event swimming question tonight

18. Aditya Bhasin
Posted September 4, 2019 at 1:38 pm | Permalink | Reply

the first race straight forward A finished before B who in turn finished before C. Hence A.B.C one after the other. Then B was not first and D was not last. E finished two places behind C Hence ABC-E the Q says D finished in a better position in 2nd race than 1st race D is not first in 1st race and will be after C the info doesn’t say D was before B and C but a better position in 2nd than 1st means they could not be 1st in the 1st hence ABCDE.
The next race

19. Aditya Bhasin
Posted September 4, 2019 at 1:49 pm | Permalink | Reply

the next race same kind of info. E did not win the 2nd race and A finished before B who again in turn is before C hence ABC in some order D finished in a better position in 2nd than first hence was not 4th or 5th. and finished before B A wasn’t 2nd and also was before B. A finished in a better position in 4th than in second race hence could not have won 2nd race as no position is better than first. as they are not 2nd also they will be 3rd and before B hence ABC. then D is not 4th or fifth so D won and E before A as A was not 2nd also.

20. Aditya Bhasin
Posted September 4, 2019 at 1:50 pm | Permalink | Reply

DEABC

21. Posted September 6, 2019 at 10:28 am | Permalink | Reply

Hello Aditya: I am working on it now in my coffee time (again). Another doubt/clarification I need is: Can one country get more than one medal in any one event? I think in both the cases, if we assume so, and if we do not assume so — the final matrix results will be v different…

22. Posted September 6, 2019 at 10:41 am | Permalink | Reply

Oops..I see…it is possible that a country can win all medals in any one event; it is already a part of the data that USSR was the only country to win all three medals in one event…

23. Aditya Bhasin
Posted September 6, 2019 at 12:34 pm | Permalink | Reply

Hi Nalin, Yes it is possible for e.g. GB won a silver and bronze in one event (a single event not separate) and the last line says that GDR won a silver and bronze also in one 100 m event. The matrix I have given you is the actual answer from the book but I was only able to do a part of it (the breaststroke events), the gold medals USA won also I deciphered would be 100 M freestyle and 100 M backstroke as they did not win “any” butterfly events (neither 100 M or 200 M) or any 200 M event. This part I did was correct, I also saw that GDR would win 100 M butterfly as no other country was mentioned USA won 2 gold in freestyle, backstroke, G Britain won the breast stroke and the reason for this is Because it says in the Question that USSR won a silver “in the event won” by GB which was not freestyle, it also says USSR did not win any backstroke event. And the last info. I supplied was that USSR did not win any medal in 100 M butterfly which leaves breaststroke.
I saw that GDR won a silver in breastroke event which could be 100M or 200M but because USSR won the silver in 100M the only other event is the 200 M and as it says in the Question that GB “won” a 200 M “event” in which the GDR had no medals hence GB can not win gold and because they won 2 medals in Breaststroke events it would be Bronze and there only bronze in 200 M.
This part was correct, Nalin I wrote to you just to discuss and clarify if I am not making any mistake or if I have deciphered the Question correctly. I have many other puzzles to share mostly from English authors some that I need similar help I would like to share to gain another point of view or if we both agree.
The rest of the Q I did not copy but also saw that GDR would be gold in 100m butterfly but not silver as they did not win gold and silver in same event. hence USSR could not win and neither could USA both did not win butterfly only silver would be GB and bronze also. And USA won both the gold in freestyle and backstroke and USSR won the bronze in 100m freestyle but GDR did not win silver here because they already have a silver in 200 M breaststroke and they need to have won a silver and bronze in the same event. Then I inferred that USSR won the 200 M breaststroke because GDR had silver and GB bronze gold could not be GDR and hence was USSR. Sorry I tend to write a lot, this is all the info that I inferred it matched the answer but needed some clarification to see if any one would agree and need to discuss my other puzzles.

Regards Aditya

• Posted September 6, 2019 at 12:55 pm | Permalink | Reply

Hello Aditya: To be frank, my first attempt was a wrong solution. This problem certainly merits more effort and is more difficult than I thought initially. I will work on it tonight again. BTW, my two cents is worth…as a general rule, never look up the solution of a math problem or logical puzzle; it will stifle your mathematical problem-solving skills/stamina…Some times, rather than working on such extremely difficult puzzles for a long duration, it is better to find a technical problem or in your case, a logic puzzle or even a Mensa puzzle, which you feel is less difficult than the current problem one is breaking one’s head with. I have seen that I have been able to crack v v difficult math problems of famous contests like William Lowell Putnam or INMO/RMO after a break of 2-3 days !! That is the subconscious mind working….try it…

24. Aditya Bhasin
Posted September 6, 2019 at 12:39 pm | Permalink | Reply

* USSR won silver in the “100M” event which was not freestyle.

25. Posted September 6, 2019 at 2:30 pm | Permalink | Reply

Hello Aditya: I strongly recommend that you buy the following book: How to Solve It: A new aspect of mathematical method by George Polya. It is v v v cheap and available on Amazon India.

Yes, I would be glad to discuss your solutions and compare my solutions with yours — for all such puzzling matters !! You are most welcome to continue in this thread in itself.

BTW, I have yet to read your solutions to the first puzzle you posed; it is generally not proper to comment unless one has devoted sufficient time to thinking about the solution on one’s own — in all such puzzles or logical/analytical questions of various olympiads or Mensa etc. Let me once again attack the puzzle with a fresh approach — then I will comment on your approach. Regards NP

26. Posted September 6, 2019 at 3:48 pm | Permalink | Reply

Hello Aditya — (a) So you have not copied the full text of your question initially itself ? ?? (b) But even if you did not, I would like to comment that your reasoning more or less is similar to mine. Especially, in my attempt today after I had started with a fundamental assumption that “GBR (UK) won a total of 2 medals in breaststroke events, so I assumed that both could be gold and so according to my fundamental assumption UK won 100 m breaststroke event and 200 m breaststroke event. I then tried to fill in the matrix but I found there were too many unknowns so I made a second assumption that UK (GBR) won a silver medal in 100 m backstroke and bronze in medal in 100 m butterfly. Then, again I tried to “fill in” the matrix…so this is a general way to proceed. What you are writing for me are detailed clarifications…Do not worry your approach is quite right…

Feel free to discuss the next new question with me — but please write the full text of the question for meaningful dialogue/interaction.

27. Posted September 6, 2019 at 7:25 pm | Permalink | Reply

Hello Aditya Bhasin: I have a little idea…just bouncing off…you can download a free software called Freemind…it will help you “systematize” thought processes in solving such huge “matrix” kind of problems….It is quite user friendly…It can be used by non-technical people also …You may want to give it a shot …regards. PS: I think I am going to try it with such problems when I have some leisure time…

28. Aditya Bhasin
Posted September 15, 2019 at 8:47 am | Permalink | Reply

Hi Nalin,

Sorry I’ve been a little busy will post some other queries but will be busy for another 5-6 weeks as my Mum is visiting me Will post when she leaves.

29. Posted September 15, 2019 at 9:44 am | Permalink | Reply

Okay regards

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