This month’s Insights puzzle arises out of a peculiar weekly bonding ritual of four fair-minded friends. Amar, Bob, Ching and Danica are math graduate students who share a house. Their living room has a fabulous rug that they call the “magic carpet,” shown above, around which they gather every weekend for a feast. The four have strong likes and dislikes, so they have instituted rules to ensure that everyone enjoys the get-togethers and can bond with each other member of the group. At every feast they have four “featured items” for the evening related to food, drink, activity or entertainment. The rules are:

- Each person must like at least half of the featured items.
- For exactly half of the items, each pair of friends must share either a like or dislike for those items.

As Bob entered the living room for a recent get-together, he wondered what the items would be. They were going to have a rice entrée, ice cream for dessert, followed by coffee and a game of Sudoku. “Had Amar decided the items for today?” Bob wondered. “It had to be him: He likes all this stuff.” As he took his place around the mat, rice soup in hand, Bob thought, “I guess I can suffer rice for an evening.” After all, it was mathematically guaranteed to be a pleasant evening, as Amar had followed the rules. Each one of them had at least two items that they liked. Each pair of friends had a couple of items for which their like or dislike coincided and over which they could enthuse or commiserate. The pairwise bonding situation was neat and balanced. Just like Sudoku.

Question 1:

Can you figure out the likes and dislikes of each friend? Write them out as a Sudoku square with the friends’ names alphabetically along the rows, and the items, in the order given, along the columns. (

Update:The solution is now available here.)

Question 2:

On a special occasion, the four friends decided to invite two of their colleagues from next door, Emma and Fred. But try as they might, they found they could not find a way to make their rules fit six people. They soon realized, however, that if they invited two more friends, Gary and Hannah, they could make things work in exactly the same way. The eight of them had a great evening, adding jazz, pizza, movie talk and liqueur to their four previous items. At this grand get-together, each pair of people had some items for which their likes or dislikes coincided and the total number of such items was exactly four. Can you make a Sudoku square for this situation, as you did for Question 1?

This puzzle was suggested by the mathematician Terence Tao, who was featured in Erica Klarreich’s article *A Magical Answer to an 80-Year-Old Puzzle*. But it is more closely related to a puzzle described in her article *A Design Dilemma Solved, Minus Designs*.

Question 3:

For those of you who love word puzzles, here’s a treat for you: The answers to this month’s puzzles are examples of mathematical structures that are known by a two-word name. The letters spelling out these two words are embedded in the text of today’s column. Can you find them?

Finally, we have to solve the mystery of the magic carpet.

Question 4:

What do you think the fabulous rug represents?

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Solve a variation of Thomas Kirkman’s puzzle by arranging nine girls in walking groups. And think fast — the clock is ticking.

Happy puzzling, and may the insight strike you!

*Editor’s notes: The reader who submits the most interesting, creative or insightful solution (as judged by the columnist) in the comments section will receive a* Quanta Magazine* T-shirt. ( Update: The solution is now available here.)*

*If you’d like to suggest a favorite puzzle for a future Insights column, submit it as a comment below, clearly marked “NEW PUZZLE SUGGESTION” (it will not appear online, so solutions to the puzzle questions above should be submitted separately).*

*Note that we will hold comments for the first day to allow for independent contributions.*

To make it easier on myself, 1 = Like, 0 = Dislike

#1)

Given:

Amar likes everything

Bob dislikes rice

everyone must like at least half the items

each pair of people will agree on two items and disagree on two items

the entries we can start with on the square are then

A 1 1 1 1

B 0

C

D

to make sure each pair agrees on 2 items and disagrees on the other 2, B C and D each have to like 2 items and dislike 2 items so that their pairings with A fit the rule. Also, this will ensure the other rule is followed. There are plenty of combinations to make sure the rules are followed.

A 1 1 1 1

B 0 0 1 1

C 0 1 0 1

D 0 1 1 0

Any combination where BCD all share exactly one dislike and each of them has 2 likes and dislikes will suffice as well.

#2)

A 1 1 1 1 1 1 0 0

B 1 1 1 1 0 0 1 1

C 1 1 0 0 1 1 1 1

D 0 0 1 1 1 1 1 1

E 0 1 0 1 0 1 0 1

F 0 1 0 1 1 0 1 0

G 1 0 0 1 0 1 1 0

H 1 0 0 1 1 0 0 1

#3) no idea

#4)

The carpet represents a simple solution for everyone's likes and dislikes to satisfy the rules if we hadn't been told that Amar liked everything. The squares with lines going bottom left to top right are likes. The squares with lines going from top left to bottom right are dislikes. The top row is Amar, second row is Bob, etc. The first column is the first featured item, the second column is the second item, etc. The rules are satisfied because each person likes 3 of the 4 items, and each pair shares a like for two of the items.

1) The solution I get is the same as Michael and in alphabetical order

Amar: 1,1,1,1

Bob: -1,-1,1,1

Ching: 1,-1,1,-1

Danica: -1,1,1,-1

2) The solution for eight can be generated from four as

S4 S4

-S4 S4

This way once again, half the columns in every two rows match and half do not match

1,1,1,1,1,1,1,1

-1,-1,1,1,-1,-1,1,1

1,-1,1,-1,1,-1,1,-1

-1,1,1,-1,-1,1,1,-1

-1,-1,-1,-1,1,1,1,1

1,1,-1,-1,-1,-1,1,1

-1,1,-1,1,1,-1,1,-1

1,-1,-1,1,-1,1,1,-1

3) The mathematical structures are known as Hadamard Matrices. The sentence "Had Amar decided.." contains Hadamard and "sat around the mat, rice soup …" has the word matrices. These matrices have for any two rows, half the columns with same entries and another half with divergent entries so that any two rows are orthogonal. After 1, 2; the order of any Hadamard Matrix is a multiple of 4.

4) The rug is a very nice solution to the 4 people puzzle

Answer to question 1. borrowing the representation. 1= like, 0= dislike

R I C S TOTAL

A 1 1 0 1 3

B 0 1 1 1 3

C 1 0 1 1 3

D 1 1 1 0 3

AB 0 1 0 1 2

AC 1 0 0 1 2

AD 1 1 0 0 2

BC 0 0 1 1 2

BD 0 1 1 0 2

CD 1 0 1 0 2