Jul 23

## Piece of Cake

In how many equal pieces can you cut a round cake using only 3 slices?

SOLUTION

Eight pieces. Cut the cake into four identical pieces with two vertical slices and then make a third horizontal slice through the center.

Jul 21

## Black and White

A boy draws 2015 unit squares on a piece of paper, all oriented the same way, possibly overlapping each other. Then the colors the resulting picture in black and white chess-wise, such that any area belonging to an even number of squares is painted white and any area belonging to an odd number of squares is painted black.

Prove that the total area of all black parts is at least one.

Draw a grid in the plane which is parallel to the sides of the squares. Then take the content of each cell of the grid and translate it (move it) to some chosen unit square. The points in that unit square which are covered by odd number of black pieces color in black, the rest – in white. It is easy to see after doing this, the entire unit square will be colored in black (each of the 2015 squares covers it once completely). This implies that the total area of black pieces is no less than 1.

Jul 21

## Ring 2

Cast Ring 2 is the second “ring” puzzle from Hanayama, coming several years after the company introduced the original Cast Ring. Even though the new one is expected to be an improvement over the previous puzzle, it is in fact inferior in almost every way. The first thing which grabs attention is how small and flimsy the toy is – looks more like a trinket than a high-quality product. It also feels cheap and hardly stays put together. Its difficulty is supposed to be 5 – compared to 4 for the old model, but actually is considerably easier. So far Ring 2 is my least favorite Hanayama puzzle and I really don’t recommend you buying it, unless you want to fully complete your collection.

• difficulty 5/6
• piece arrangement
• put together

Jul 21

In the Padurea forest there are 100 rest stops. There are 1000 trails, each connecting a pair of rest stops. Each trail has some particular level of difficulty with no two trails having the same difficulty. An intrepid hiker, Sendeirismo has decided to spend a vacation by taking a hike consisting of 20 trails of ever increasing difficulty.
Can he be sure that it can be done?

He is free to choose the starting rest stop and the 20 trails from a sequence where the start of one trail is the end of a previous one.

Place one hiker in each of the rest stops. Now, go through the trails in the forest one by one, in increasing difficulty, and every time you pick a trail, let the two hikers in its ends change places. This way the 100 hikers would traverse 2000 trails in total, and therefore one of them would traverse at least 20 trails.

Jul 20

## Merlin and Hermes: Mysterious Lines

Two adventurers, Merlin and Hermes, approached a large iron door built into a cliff face.”Well…”, said Hermes, “What do we do now?”. Merlin produced an old, large piece of crumpled paper from his pocket. “Hrm…”, Merlin mumbled. “It says here that we must speak the six letter keyword to open the door and enter the secret chamber, but I don’t remember seeing any signs as to what that keyword might be…”

After a bit of searching, Hermes notices something etched into the ground. “Come over here!”, he yelled, pointing frantically. And sure enough, barely visible and obscured by dust, was a series of lines of differing color etched into the ground:

“Ah”, Merlin said, “So that is the keyword.” Hermes was lost and confused. After staring at it for another thirty seconds, he grumbled “What keyword!? All I see is a bunch of lines!”. Merlin simply responded, “You’re just looking at it the wrong way. It’s obvious!”

Isn’t it?

The signs are engraved letters on the ground and Merlin and Hermes are looking at them from above (the italic “looking at it the wrong way” is a hint). The darker a part from some sign is, the farther from the ground it is. The only letters which could correspond to this description are U – N – L – I – N – K. Therefore the keyword is “UNLINK”.

Jul 20

## Perplexus Original

Perplexus Original is the very first Perplexus maze created and probably the most popular one. It is rated with medium difficulty and this seems fair. Even though Perplexus Original supposedly has 100 barriers, most of them are just simple curves and drops, requiring no effort to pass. Ultimately, there are no more than 15 places along the entire track which cause difficulties, but they are still enough to pose a good challenge. In case you don’t want to start every time from the beginning of the maze – there are 2 alternative starting points in it. This allows you to place the ball at the 1st, 26th or 59th barrier, skipping all of the previous ones. The design is very good, using bright colors, making it easy to follow the track. There are several barriers which are lots of fun to pass through, such as the big spiral in the middle and few moving obstacles around. The overall construction is solid as well.

Even though Perplexus can not be called a “puzzle” in the real sense of the word, it still requires good hand-eye coordination, movement precision and intense focus. It is a great toy to have at home, especially if you have children around.

• medium difficulty
• 100 barriers

Jul 17

## Battleship

A battleship starts moving at 12 PM from an integer point on the real line with constant speed, landing on every hour again on an integer point. Every day at midnight you can shoot at an arbitrary point on the real plane, trying to destroy the battleship. Can you find a strategy with which you will eventually succeed to do this?

If we know the starting point of the battleship and its speed, then we can determine its position at any time after 12 OM.

There are countably many combinations (X, Y) of starting point and speed. We can order them in the following way:

(0, 0) – starting point 0, speed 0;
(0, 1) – starting point 0, speed +1;
(1, 0) – starting point 1, speed 0;
(0, -1) – starting point 0, speed -1;
(1, 1) – starting point 1, speed +1;
(-1, 0) – starting point -1, speed 0;
(0, 2) – starting point 0, speed +2;
(1, -1) – starting point 1, speed -1;
(-1, 1) – starting point -1, speed 1;
(2, 0)- starting point 2, speed 0,
and so on. Of course, we can choose the ordering in many different ways.

Now we can start exhausting all possibilities one after another. First we assume the combination is (0, 0), calculate where the battleship would be at midnight during the first day and shoot there. Then we assume the combination is (0, 1), calculate where the battleship would be at midnight during the second day and shoot there. If we continue like this, eventually we will hit the battleship.