# Lenny’s Logical Internship

Lenny recently applied for an internship at Logicorp™, a company built around using logical reasoning to solve any sort of problems their clients face. The interview was long and difficult, and Lenny was asked many personal questions ranging from his favorite mathematician to his favorite color, and was given multiple logic tests. Considering that Lenny is a perfect logician, he aced the interview and was quickly offered a job. Today is his first day. After showing up to work bright and early, Lenny was given his first task: sorting paperwork.

Lenny’s manager handed him a box containing 9 folders of papers, corresponding to 9 of Lenny’s co-workers: Aristotle, Boole, Cantor, DeMorgan, Euclid, Fourier, Gauss, Hopper, and Ockham. He handed Lenny a ream of nine colors of stickers: Red, Orange, Yellow, Green, Blue, Purple, Black, White, and Gray. He asked Lenny to label each of the folders with a different color sticker. He then pointed Lenny to a set of three cabinets, each with three shelves, and instructed Lenny to file each of the folders into a different cabinet shelf. Before Lenny could protest that he did not know which color went with which file, nor where each file should go, his manager handed him a sheet of paper with the following instructions on it:

Each of the files must be labeled with a unique color and placed into a unique shelf, fulfilling the following requirements:

1. Each cabinet1 should contain exactly one primary color2.
2. Mr. Boole’s and Mr. DeMorgan’s files should be labeled with shades of gray3.
3. The files labeled with the Red, White, and Blue stickers should belong to employees whose initials4 are alphabetically consecutive.
4. Employees with a vowel initial should have files in the first cabinet.
5. Files in the middle shelves should not be labeled with warm colors5.
6. Employee’s files should not be labeled with colors that contain their initial (U.S. spelling).
7. Colors formed by mixing two other colors must be filed adjacent6 to the colors that formed them7.
8. The middle shelf of the middle cabinet should contain the files of an employee whose initial is one of the first five letters of the alphabet.
9. Mr. Fourier’s file should not be placed in the bottom row.
10. You should label the file in the bottom shelf of the third cabinet with your favorite color.

Clarifications:
1 A cabinet is a single column of three shelves
2 The primary colors are red, blue, and yellow
3 The shades of gray are white, gray, and black
4 The employee’s initials are A, B, C, D, E, F, G, H, and O (they each have only one)
5 The warm colors are red, orange, and yellow
6 Orthogonally adjacent (above or below in the same cabinet / to the right or left in the same row)
7 Red and blue mix to form purple; red and yellow to form orange; yellow and blue to form green; black and white to form gray. Assume no other colors can be formed by mixing

Lenny understood that this was a test of his logical wit. After studying the rules and making a few important deductions, he confidently placed a single sticker on each folder, filed each folder into a separate shelf, and returned to his manager. His manager was quite impressed that Lenny had correctly deduced the correct pairings of files, colors, and locations without any guesswork.

Now it’s your turn to apply for Logicorp™. Can you figure out how to file everything? Remember: There is only one solution, which Lenny was able to reach through pure logical deduction.

Here’s a visualization of how the cabinets are laid out:

Answers should show the color corresponding to each employee’s file and where that file goes, along with the logical steps one took to come to that conclusion.

Explanation below. Let me know if anything doesn't look right. I'll call all the characters by their first initial for ease.

1. First, notice that all colors have E except for Black and Gray. E is a vowel so E will be in the first column along with A and O. This means that E cannot be Gray. If E were Gray then it would have to be adjacent to Black and White which are B and D, but since it is in the first column with A and O that is not possible. Thus, E is Black.

2. The colors mixing rule is surprisingly strict. Let's look at the colors in two groups: grays and rainbows. We know from above that the grays form either a horizontal line or an 'L' shape with one leg in the first column. Next, notice that the rainbows all must be connected to two other rainbows. The primaries are all connected to the colors they form, and the mixed colors are connected to the two primaries that make them up. This means that the grays must be a horizontal line, since if it is an 'L' it leaves squares that can only make one connection, but we know all rainbows connect twice. Further, since the rainbows are interconnected, the grays must be either the top horizontal or the bottom horizontal. We also know that the warm colors cannot be in the middle. So Red, Orange, Yellow will be either in the top horizontal or bottom, and Green, Blue, Purple will be in the middle horizontal. Blue must be inbetween the other two so Blue is in the middle cabinet, middle row. However, that square can't be A, E because they are in the first column, or B, D, because it is not Gray. So the middle cabinet, middle row is Blue and C.

3. Red, White, and Blue have alphabetically initials. Blue is C so White is B and Red is A.

4. Red is A tells us that Red is in the first column. That means that Purple is in the first column, in fact it is first column second row, by the ideas above. This means that Green is third column second row. This also means that Purple is the last one in the first column we don't know, so it is O.

5. The last gray is Gray and is D.

6. There are only Orange, Yellow, and Green left unidentified as either F, G, and H. Orange and Green have G so G is Yellow.

7. Now, we are here. . There are two possibilities. Either the warms are on the top row, or the bottom row. We can fill out both and the compare, we get Possibility 1. Possibility 2.

8. Finally, we know Lenny can fully determine this with logic and we haven't used Rule 9. Rule 9 is no helpful in Possibility 2, and actually both solutions in Possibility 2 should fulfill all Rules. However, then Lenny would not have been able to know without guessing. Thus, we conclude that Lenny's favorite color is Yellow and we are in Possibility 1. No, we can use Rule 9 to fill out the final two color/person pairs and finish the puzzle.

End Note.

Rule 1 is fulfilled, but I don't think I had to use it.

• Nice work! rot13(Tbbq pngpu ba abg arrqvat Ehyr 1. V guvax V pbhyq unir jbeqrq Ehyr 3 n yvggyr orggre nf gur vagragvba jnf abg gung gur erq, juvgr, naq oyhr pbybef unq gb or va gung beqre pbafrphgviryl (vg pbhyq unir orra oyhr, juvgr, erq sbe rknzcyr); vg whfg jbexrq bhg gung znxvat gung nffhzcgvba fgvyy yrq gb gur pbeerpg nafjre.) Feb 6, 2020 at 14:16
• Ah, I see. I was a bit confused on that wording, but this seemed to work. That makes sense though. Nice puzzle!
– cmxu
Feb 6, 2020 at 14:42
• I'm wondering if there's a better way to word that rule, or if I should have just given examples. At least it worked out Feb 6, 2020 at 14:47
• Not sure. I do think an example may have helped to clarify things though.
– cmxu
Feb 6, 2020 at 14:55