Puzzle Tournament 4

Time for work: 1 hour

Each problem is worth 1 point. Use the form at the bottom of the post to send your solutions.

1. The Grid

by Puzzle Prime

Figure out how the last portion (7×5 in yellow) of the grid should be colored in black and white.

2. Hexado

by Dr. DJ Upton

Place arrows along hexagon edges so that the number of arrows pointing to each hexagon equals the number of dots inside, adhering to the following rules:

  1. Arrows cannot be touching.
  2. Arrows cannot be placed on dashed edges.

3. Segments

by Puzzle Prime

Use at most 27 segments to create the largest number with distinct digits.

Notes: For example, the number 273914 would use 5+3+5+6+2+4=25 segments.

4. Constellations

by Raindrinker

Connect the stars with lines, so that the number inside each star corresponds to the number of lines connected to it, and the number outside each star corresponds to the total number of stars in its group.

Note: No line connecting two stars can pass through a third star.


5. Chess Connect

by Puzzle Prime

The starting and ending positions of 6 chess pieces are shown on the board. Find the trajectories of the pieces, if you know that they do not overlap and completely cover the board.

Notes: The pieces can not backtrack. Two trajectories can intersect diagonally but can not pass through the same square. Only the Knight has a discontinuous trajectory.


6. Broken Square

by Puzzle Prime

Use exactly 5 out of these 16 pieces to build a 7×7 grid, without overlapping.

Note: You can rotate the pieces, but you cannot mirror them.

We Go on Vacation

For this puzzle/game, you will need to find a group of friends, preferably 5 or more. The premise is that you will go together on a vacation but each of you can bring only specific items there. The rules regarding which items can be brought and which not are known by one of the players and the other ones are trying to guess them.

In the exchange below, George is the one organizing the trip and the one who knows which items the rest are allowed to bring.

GEORGE: I will take my guitar with me. What do you want to take?

SAM: Can I take an umbrella with me?

GEORGE: No, you cannot take an umbrella, but you can take some sunscreen.

HELLEN: Can I take a scarf with me?

GEORGE: No, you cannot take a scarf, but you can take a hat.

MONICA: Can I take a dress with me?

GEORGE: No, you cannot take a dress, but you can take some makeup.

Can you guess what the rules of the game are?

Everyone is allowed to take with themselves only items whose first letter is the same as the first letter of their names. Thus, George can take a Guitar, Sam can take Sunscreen, Hellen can take a Hat, and Monica can take Makeup.

Leave No Squares

How many matchsticks do you need to remove so that no squares of any size remain?

Nine matchsticks are enough, as seen from the solution below.

To see that eight matchsticks are not enough, notice that removing an inner matchstick reduces the number of 1×1 squares at most by 2. Since there are 16 such small squares, in order to get rid of them all, we need to remove only inner matchsticks. However, in this case, the large 4×4 square will remain.

The Die Game

You pick a number between 1 and 6 and keep throwing a die until you get it. Does it matter which number you pick for maximizing the total sum of the numbers in the resulting sequence?

In the example below, the picked number is 6 and the total sum of the numbers in the resulting sequence is 35.

No matter what number you pick, the expected value of each throw is the average of the numbers from 1 to 6 which is 3.5. The choice of the number also does not affect the odds for the number of throws until the game ends, which is 6. Therefore, the total sum is always 3.5 × 6 = 21 on average, regardless of the chosen number.