Tanya Khovanova's Math Blog

The second “instructioned” puzzle is Portals by Palmer Mebane. It is an insanely beautiful and difficult logic puzzle that consists of known puzzle types interconnected to each other through portals. Here Palmer Mebane explains how portals work:

“Each of the ten puzzles corresponds to a color, seen above the grid where the name of the puzzle is written. The grid contains nine square areas, one each of the other nine colors. These are portals that connect the puzzle to one of the other nine, as denoted by the portal color. Each puzzle’s rules define which squares of their solution are “black”. On the portal squares, the two puzzles must agree on which squares are black and which are not. For instance, if in the red grid the top left square of the blue portal is black, then in the blue grid the top left square of the red portal must also be black, and vice versa.”

On the Portals puzzle page you can find the rules for how each individual game is played and how to shade areas. The puzzle requires a lot of attention. It took us a long time to test-solve it. If you make a mistake in one grid it will propagate and will lead to a contradiction in another grid, so it is difficult to correct mistakes. If you do make a mistake, you are not alone: we kept making mistakes during our test-solve. Because of the difficulty of tracing back to the source of the error, we just started anew, but this time making sure that every step was confirmed by two people. Working together in this way, we were able to finish it.

If you do not care about the extraction and the answer, ignore the letter grid in the middle and enjoy the logic of it.

There were a couple of puzzles during the MIT Mystery Hunt that were not so mysterious. Unlike in traditional hunt puzzles, these puzzles were accompanied by instructions. As a result you can dive in and just enjoy solving the logic part of the puzzle without bothering about the final phase, called the extraction, where you need to produce the answer.

The first puzzle with instructions is Random Walk by Jeremy Sawicki. I greatly enjoyed solving it. In each maze, the goal is to find a path from start to finish, moving horizontally and vertically from one square to the next. The numbers indicate how many squares in each row and column the path passes through. There are nine mazes in the puzzle of increasing difficulty. I am copying here two such mazes: the easiest and the toughest. The colored polyomino shapes are needed for the extraction, so you can ignore them here.

Today I have a special treat for you. Here is the first of several puzzles that I plan to present from the 150 that we used in the MIT Mystery Hunt 2013. Keep in mind that although the puzzles have authors, they were the result of a collaboration of all the team members. In many instances editors, test-solvers and fact-checkers suggested good ways to improve the puzzles.

I wrote the puzzle Open Secrets jointly with Rob Speer. The puzzle was in the opening round, which means it is not too difficult. By agreement the answers to the puzzles are words or phrases. I invite my readers to try this puzzle. I will post the explanation in about two weeks.

The father of my son has four children. My son is my only child. How many children do we have in total together?

I discovered the following chess puzzle on a Russian blog for puzzle lovers. It is a helpmate-type puzzle. Black cooperates with White in checkmating himself. In this particular puzzle Black starts and helps White to win in one move.

Oops. Something is not quite right. There are not enough pieces on the board. To recover the missing pieces in order to solve the puzzle, you need to retrace your steps. If Black and White go back one move each, they will be able to cooperatively checkmate Black in one move. Find the position one move back and the cooperative checkmate.

For my every class I try to prepare a challenge problem to stretch the minds of my students. Here is a problem I took from Adam A. Castello’s website:

There is a ceiling a hundred feet above you that extends for- ever, and hanging from it side-by-side are two golden ropes, each a hundred feet long. You have a knife, and would like to steal as much of the golden ropes as you can. You are able to climb ropes, but not survive falls. How much golden rope can you get away with, and how? Assume you have as many hands as you like.

The next problem I heard from my son Sergei:

You are sitting at the equator and you have three planes. You would like to fly around the equator. Each plane is full of gas and each has enough gas to take you half way around. Planes can transfer gas between themselves mid-air. You have friends, so that you can fly more than one plane at once. How do you fly around the equator?

In what base does 2012! have more trailing zeros: base 15 or 16?

Explain why the result shouldn’t be too surprising.

Thank you to everyone who helped me to find a host for my Number Gossip website. Some readers and friends even offered me free hosting on their servers. I decided to pay for hosting because I have many specific requirements and that might be a burden on my friends.

On the basis of my readers’ recommendations, I chose Dreamhost as my new webhosting provider. I apologize for the interruption in the flow of the gossip. I know that many people use Number Gossip for birthday gift ideas. I can tell you that on my previous birthday, you could have congratulated me on becoming prime and evil.

Consider central symmetry: squares and circles are centrally symmetric, while trapezoids and triangles are not. But if you have two trapezoids, which of them is more centrally symmetric? Can we assign a number to describe how symmetric a shape is?

Here is what I suggest. Given a shape A, find a centrally symmetric shape B of the largest area that fits inside. Then the measure of central symmetry is the ratio of volumes: B/A. For centrally symmetric figures the ratio is 1, and otherwise it is a positive number less than 1.

The measure of symmetry is positive. But how close to 0 can it be? The picture on the left is a shape that consists of five small disks located at the vertices of a regular pentagon. If the disks are small enough than the largest symmetric subshape consists of two disks. Thus the measure of symmetry for this shape is 2/5. If we replace a pentagon with a regular polygon with a large odd number of sides, we can get very close to 0.

What about convex figures? Kovner’s theorem states that every convex shape of area 1 contains a centrally symmetric shape of area at least 2/3. It is equal to 2/3 only if the original shape is a triangle. That means every convex shape is at least 2/3 centrally symmetric. It also means that the triangle is the least centrally symmetric convex figure. By the way, a convex shape can have only one center of symmetry.

After I started writing this I discovered that there are many ways in which people define measures of symmetry. The one I have defined here is called Kovner-Besicovitch measure. The good news is that the triangle is the least symmetric planar convex shape with respect to all of these different measures.

I received the book Taking Sudoku Seriously by by Jason Rosenhouse and Laura Taalman for review and put it aside to collect some dust. You see, I have solved too many Sudokus in my life. The idea of solving another one made me barf. Besides, I thought I knew all there is to know about the mathematics of Sudoku.

One day out of politeness or guilt I opened the book — and couldn’t stop reading.

The book is written for people who like Sudoku, but hate math. This is so strange. Sudoku is math. People who are good at Sudoku are good at math, or at least they are supposed to be. It seems that math education in the United States is so bad that people who were born to be good at math and to like math, hate it instead. So the goal of the book is to establish a bridge from Sudoku to math. And the book does a superb job of it.

This well-written book moves from puzzles to discussions in such a natural way that math becomes a continuation of puzzles.

Taking Sudoku Seriously covers a lot of fun material: methods to solve Sudoku, how to count the number of different Sudoku puzzles, and how to find the smallest number of clues that are needed for a unique puzzle. The book travels into the neighboring area of Latin and Greco-Latin squares. While discussing all those fun things it covers groups, symmetries, number theory, graph theory (including book thickness) and more.

I am not the target audience for this book, because I do not need convincing that math is fun. The best part for me was the hundred puzzles. Only a portion of them were standard Sudoku puzzles — and I skipped those. The others were either Sudoku with a twist or plain math puzzles.

The puzzles are all very different and I was so excited by them, that I went ahead and solved them, and caught up with reading the text later. And I enjoyed both: reading and solving.

Here is puzzle 91 from the book. Fill in the grid so that every row, column, and block contains 1-9 exactly once. In addition, each worm must contain entries that increase from tail to head. For blue worms you must figure out yourself which end is the head.