Test Grades Logic Puzzle SOLUTION

This is the solution to my logic puzzle, "Test Grades Logic Puzzle".

TIPS: 

-Draw yourself a grid! Work this puzzle like a Sudoku and write in the corner of each cell what possible grade each student could have received. Eliminate the possibilities as more clues come out.

-Make sure you have written down which letter goes where. It is very unlikely you will complete the quiz in 20 minutes, so when it times out and you need to restart, you will be easily able to enter all the grades you've worked out so far.

-Always bear in mind all the previous clues you've received, as they can become important later.

-When you've reached a certain point and the quiz has timed out, work out the next hint before restarting the quiz.

If you don't want the exact answers, do not read further.

STEP 1 - Inferring Seat 09. Seat 01 tells you off the bat what seat 09 got. They were 1% off an A. So we know instantly seat 09 got a B.

STEP 2 - Inferring Seat 07. Seat 09 tells you that someone in their row got an A. Seat 01 tells you that all the people who got A's are prime numbered seats (but not 02, 03, or 05). The only prime number in seat 09's row is 07, therefore 07 got an A.

Seat 07 has some very important clues which will become useful later.

STEP 3 - Inferring Seat 02. Seat 07 tells us the person in front of him got an F. That is seat 02.

So far so easy. This is where the difficulty starts to ramp up.

STEP 4 - Inferring Seats 23 and 29. Seat 02 is lying to us, we know that, as they got an F. As 02 tells us there is a row without an A, we know there must be at least one A in every row. Seat 01 has already told us there are 7 A's in the class, and that all the rest are prime numbers but not 02, 03 or 05. This leaves 6 A's in 7 remaining prime numbers: 11, 13, 17, 19, 23, and 29. One of these is not an A. Seats 23 and 29 are the only prime numbers on their respective rows, and as there must be an A in every row, these two seats must both have an A.

Bear in mind seat 23 tells us the configuration of A-B-C-D-F is a palindrome. This tells us there are 7 F's.

STEP 5 - Inferring Seat 04. Seat 29 tells us three seats in their column are lying, but calls out seat 04. With the information we have right now, we know the liars have D or F, and maybe C. But 23 tells us there are no C's in this column. So seats 04, 14 and 24 are all D or F. We know from seat 07 that there is only one F in the front row, and that's 02. Therefore, 04 can only be a D.

STEP 6 - Inferring Seats 06, 10 and 12. Seat 04's lying hint informs us that 03 and 08 got better than a D. Remember that seat 09 has told us two people on the 2nd row got an F. We now know that seat 08 did not get an F, leaving seats 06 and 10 as the two in the row who got an F. Not only that, seat 07 has told us that three of the four people surrounding them (no diagonals) got an F. Since seat 08 didn't get an F, and we know 02 and 06 got an F, the final F surrounding 07 is seat 12.

We don't yet know what seat 08 got, but that's fine. We just know they didn't get an F.

Seat 12 also gave us some very important information: every column and row MUST have at least one repeated grade.

STEP 7 - Inferring Seats 11, 13 and 19. In one fell swoop, we can determine the rest of the A's. This is because seat 10 tells us, by lying, that seat 15 did better than seat 17. We don't even need to know what either 15 or 17 got. We just know that 15 did better than 17. This means seat 17 couldn't have gotten an A, as you can't do better than an A. This means that the remaining three valid prime numbers, 11, 13, and 19, must all be an A.

STEP 7.5 - Inferring the Distribution of Grades in the Class. Seat 11 tells us one more student got a C than a B. If we combine this with seat 23's hint about the grade distribution being a palindrome, this gives us a set of simultaneous equations. We know that there are 7 A's and F's, and that B's and D's are equal. 7+B+C+D+7 = 30. Therefore, B+C+D = 16. As B = D, 2B + C = 16. We now know that C-B = 1. By solving the simultaneous equations, we find that B=D=5, and C=6.

Therefore, the A-B-C-D-F distribution is 7-5-6-5-7. This will be very important later on.

STEP 8 - Inferring Seat 08. Seat 19 tells us that no seat that is a multiple of 4 got better than a C. Since we know from 04's hint that seat 08 did better than a D, and all the A's are now accounted for, we know 08 can only have gotten a B or a C. But seat 19's hint confirms that seat 08 got a C.

Seat 08 finally confirms to us what the situation is with the C's. One of them is lying and the rest are telling the truth. We know 08 is telling the truth, however, as they state 01 got an A, which we've known from the start.

STEP 9 - Inferring Seat 25. This one's a freebie. Seat 08 says seat 25 did better than themselves. This means 25 got a B.

STEP 10 - Inferring Seat 05. This one's a bit more challenging. Seat 25 tells us 03 and 05 didn't get the same grade, but the one who got the lower grade is lying. We now know a liar can be a C grade, too. We know nobody else on the row got an F, and we know 03 got a C or above. It's obvious 05 is the liar here. So we're left with three possibilities here. 03 and 05 respectively could've gotten: (1) B and C; (2) C and D; (3) B and D. It can't be (1) as this would mean the 01-05 row has no repeats, which we know is forbidden. We can't determine what seat 03 is, but the two remaining allowed configurations both involve seat 05 getting a D. Therefore, the only possible outcome for 05 is a D.

STEP 11 - Inferring Seat 17. We know from seat 07's clue there is still an F unaccounted for in the 2nd column (02-07-12-17-22-27). You now have enough information to allocate this missing F. Seat 08 tells us there is only one liar in seats 21-25. We know from 29's clue that this liar has to be seat 24. We therefore know that 21 and 22 are telling the truth. This means only 17 or 27 could've gotten an F. But seat 05's clue informs us seat 17 did WORSE than seat 27. You can't get worse than an F, so seat 27 did NOT get an F. Therefore, seat 17 did and we now have the third and final F in the 2nd column.

STEP 12 - Inferring Seats 16, 18 and 20. Seat 17 tells us (by lying) that both 16 and 18 the same grade, but did worse than seat 20. Remember that very useful hint seat 19 gave us: no multiple of 4 got above a C. Seat 20 got at most a C, and seats 16 and 18 did worse. We again have a few possible configurations with 16, 18 and 20: (1) D, D, C; (2) F, F, D; (3) F, F, C.

Remember, seat 02 has told us (by lying) that no row can have more than three F's. Therefore, configurations (2) and (3) are forbidden, as this would result in 16, 17 and 18 all getting an F and therefore three in a row. This means 16, 18 and 20 respectively got D, D and C.

STEP 13 - Inferring Seats 14 and 24. This is my favourite part. Seat 13 told us there is in fact a D student who is telling the truth. This is where we encounter the truthing D student. Seats 16 and 18 both accuse each other of lying, then tell us what seat 14 got, as well as seat 24's grade in relation to 14. We already know 14 and 24 are both liars and that 14 didn't get a C. But using the grade distribution, 7-5-6-5-7, we can determine which is lying. Currently, we have 4 D's and 5 F's revealed. Bear this in mind.

If seat 16 is telling the truth: seat 14 got an F and seat 24 got a D. This would reveal the 6th F and 5th D. This is fine.

If seat 18 is telling the truth: seat 14 and 24 both got a D. If this is true, there would be 6 D's revealed, which violates what we've already worked out: there are only 5 D's in the class.

We can now infer seat 18 is the liar and 16 is the D student who is telling the truth. This tells us seats 14 and 24 got an F and a D respectively.

This also shows us that 20 is actually the C student who is lying, as seat 14 (an F student, a certified liar) gives us the same clue that seat 20 does. Therefore, all C students from here on are telling the truth.

STEP 14 - Inferring Seat 28. We know by now there is one F unaccounted for still. We know from 29's hint there is one liar in the back row. We know from 06's hint a while back that there is a row without an F. This has to be the fifth row, since there is also only one liar in the fifth row, and we uncovered them - they were seat 24 who got a D. The liar in the back row must be the final F. Seat 24 tells us (by lying) the middle column has more liars than the left column. The left-most column only has one liar - 06. We know 21 isn't a liar, as its row's only liar is accounted for. The middle column also has one liar - 18. Since the middle column has more liars than the left-most column, this means the middle column must have one more unaccounted-for liar. It can't be 03, which either got a B or a C. It must be 28.

All A's, D's and F's are accounted for now. All the remaining seats either got a B or a C. With so many cells now filled in, and only B's and C's remaining, the final seven seats are MUCH easier.

STEP 15 - Inferring Seat 21. In theory, we could've inferred this seat before 28. But we know that 21 can only have gotten a B or a C, as that's all that's left at this point. Seat 14's lie tips us off that seat 21 got a different grade to 25, who got a B. Therefore, 21 got a C. Easy.

STEP 16 - Inferring Seats 15 and 30. This is perhaps the last challenging step. Seat 21 tells us three people in the right-most column got the same grade. There are two seats in the right-most row, 15 and 30, uncovered. We have uncovered two B's but we know there are 2 more in the bottom-left 2x2 grid from 28's clue. As such, if 15 and 30 both got a B, this would mean six B's, which violates the known distribution of grades. We've also uncovered 3 C's, with 1 more in the bottom-left 2x2 grid. If 15 and 30 are both C's, this would result in six C's, which adheres to the 7-5-6-5-7 distribution. Therefore, 15 and 30 both got C's.

STEP 17 - Inferring Seat 26. Since we know from 20's lie that 26 and 30 did not in fact get the same grade, we now know seat 26 got a B. 

STEP 18 - Inferring Seat 03. In the bottom-left 2x2 grid, this leaves 22 and 27 remaning, and we know 1 must be a B and 1 must be a C in that grid. Overall, there are 2 B's and 1 C remaining, so 03 must be a B.

STEP 19 - Inferring Seats 27 and 22, and WINNING. 03 tells us they did better than the "cube-numbered" seats. This is of course referring to 08 and 27. Since 03, a B, did better than 27, this shows seat 27 got a C. By sheer process of elimination, we know 22 got a B. 

And that's it! YOU WIN!!!!! Congratulations if you solved this whole puzzle without consulting the solution!! Thank you for playing! :)

FUN FACT:

When I made this quiz, I started by drawing out a 5x6 grid, told myself "7 A's, 5 B's, 6 C's, 5 D's, 7 F's", then populated the grid pretty much at random. I designed all the hints and all the clues around where the letters ended up in the grid. The ONLY change I made was switching 26 and 11 around. 26 was originally an A and 11 a B. I made this change for the sake of the "prime numbers" clue, since I realised, BY CHANCE, all the A's had landed on prime numbers except 26.