Bebras 2019 - Level 4

BeaverKing uses six types of ingredients A, B, C, D, E and F to make sandwiches. The picture below shows sandwiches and the ingredients used.



| Sandwich, left to right | Ingredients |
|---|---|
| 1 | C, F |
| 2 | A, B, E |
| 3 | B, E, F |
| 4 | B, C, D |

Which of the four sandwiches below is made from ingredients A, E and F?

A beaver has balls on three poles A, B and C as shown on the left. She can move one ball from one pole to another; each such action counts as one move.



What is the minimum number of moves needed to obtain the arrangement on the right?

Beaver Airlines has flight routes connecting cities as shown below.



To reduce environmental pollution, the airline commits to removing some routes. However, the remaining routes must still allow a beaver in any city to reach any other city. For example, if the route between San Francisco and Washington, D.C. is removed, a beaver can travel between the cities via New York. What is the maximum number of flight routes Beaver Airline can remove?

The beavers put 13 balls into a triangular box as illustrated below.



Figure label: “Hai quả bóng liền dưới” means “The two balls directly below”.

A ball is said to be in a “dangerous” position if it satisfies at least two of the following conditions:
- There is at least one empty space directly below that ball.
- There is at least one ball directly below that ball that is in a dangerous position.

How many balls in the box are not in a dangerous position?

Editorial clarification approved for grading: the original phrase “at least two” is preserved above. The intended rule in the official solution is that at least one of the two conditions suffices (OR). Answer using this intended rule. The literal AND reading is discussed separately in the solution.

A beaver is very interested in glass recycling. With the three machine types below she can transform glass as follows:

- Type 1: Two pieces of glass are needed. If both pieces are transparent, the output is one transparent piece. Otherwise the output is one coloured piece.
- Type 2: Two pieces of glass are needed. If both pieces are coloured, the output is one coloured piece. Otherwise the output is one transparent piece.
- Type 3: Only one piece is needed. A transparent input produces a coloured output; a coloured input produces a transparent output.

The beaver built the machine shown below. The Vietnamese label “Đầu ra” means “Output”.



What colours must pieces A, B, C and D have for the final output to be transparent?

Eight beavers sit around a round table. We know:
- Alice sits opposite David.
- Henry sits between Greta and Eugene.
- Franny does not sit next to Alice or David.
- One beaver sits between Greta and Clare.
- Eugene sits to David’s left.
- The remaining beaver is named Bruce.



In what order do we read their names clockwise?

There are 42 beavers sharing 5 warehouses. The first beaver uses the first warehouse, the second uses the second, …, the fifth uses warehouse 5, the sixth uses the first warehouse, …, continuing to the last beaver.

One day the sixth warehouse is completed. The beavers now decide to share the warehouses as follows. The first beaver uses warehouse 1, the second warehouse 2, …, the sixth warehouse 6, the seventh warehouse 1, …, continuing to the last beaver.



| Warehouse | First beaver numbers with 5 warehouses | First beaver numbers with 6 warehouses |
|---|---|---|
| 1 | 1,6,11,… | 1,7,13,… |
| 2 | 2,7,12,… | 2,8,14,… |
| 3 | 3,8,13,… | 3,9,15,… |
| 4 | 4,9,14,… | 4,10,16,… |
| 5 | 5,10,15,… | 5,11,17,… |
| 6 | — | 6,12,18,… |

How many beavers still use the same warehouse as before?

The magician Bibrax puts 49 coins on a table. Bibrax closes his eyes and asks a beaver to turn over any one coin. On opening his eyes, Bibrax can tell the beaver which coin was turned over. The secret of the trick is: “the number of star-side-up coins in each row and each column is initially even”.



Editable transcription (1=star, 0=blank):

| | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| 2 | 1 | 0 | 0 | 0 | 1 | 1 | 1 |
| 3 | 1 | 0 | 1 | 1 | 0 | 1 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 1 | 0 |
| 5 | 0 | 1 | 0 | 1 | 1 | 1 | 0 |
| 6 | 1 | 0 | 1 | 1 | 0 | 1 | 1 |
| 7 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |

Which coin did the beaver turn over in the picture above?

A beaver built a machine that transforms a word as follows:
- If the number of letters is even, the word is sent to the box on the right.
- If the number of letters is odd, the word is sent to the box on the left to delete the first letter. It is then sent to the right-hand box.
- In the right-hand box, adjacent pairs of letters are swapped, after which the word is sent to the last box.
- In the last box, the letters are written in reverse order.

For example, putting “CONTEST” into the machine gives “STTEON”.



Figure labels: “Có lẻ chữ cái”=odd number of letters; “Xóa chữ cái đầu”=delete first letter; “Có chẵn chữ cái”=even number of letters; “Đổi chỗ 2 kí tự liền kề”=swap each adjacent pair; “Viết ngược lại cả chữ”=reverse the whole word.

What word results when “STORAGE” is put into the machine?

Stephen the beaver makes bracelets by following the instructions below.



Each instruction on the left is replaced by the instructions on the right. The picture below shows some bracelets Stephen has made.



Translated rule labels: Bắt đầu=Start; Vòng=Ring; Nối=Join; Khóa=Clasp; Hạt=Bead; Hoa=Flower. The following editable transcription preserves their order. Let o be a ring, d a small dark bead, g a large plain bead, s a spiral bead, f a flower, and k the clasp.

| Instruction | Replacement |
|---|---|
| Start | Ring Join Clasp |
| Ring | o d o d |
| Clasp | d k |
| Join | g g Bead, or s, or Flower |
| Bead | s s Join |
| Flower | f Join f |

Which of the following bracelets was not made by Stephen?

Beaver farms are divided into small squares. The centre square is a warehouse, while the other squares are used to grow grass or wheat. Every year the beavers must report to the authorities the number of squares used to grow wheat. The report includes the total number of wheat-growing squares in each row and column, as below.



The original example labels “Cỏ” mean grass and “Lúa” mean wheat. Its editable grid is:

| Grass | Wheat | Grass |
|---|---|---|
| Wheat | Warehouse | Grass |
| Grass | Wheat | Grass |

Row totals: 1,1,1; column totals: 1,2,0.

Which of the following beaver reports is correct?

| Option | Row totals, top to bottom | Column totals, left to right |
|---|---|---|
| A | 3,0,3,2,1 | 3,0,1,1,2 |
| B | 3,2,1,2,4 | 2,2,3,2,3 |
| C | 0,3,3,1,1 | 0,3,3,1,1 |
| D | 4,1,0,3,2 | 4,1,0,3,3 |

Ten ants on island A are trying to reach island F to find food, as shown below.



At any moment only one ant can travel on a path connecting two islands. An ant takes 1 minute to traverse a whole path, and movement time on an island is negligible. What is the maximum number of ants that can reach F after 3 minutes?

Natasha the beaver has lost her robot in a park. The park is a 3×3 area as shown below. The robot may be anywhere in its 9 unit squares.



Natasha can wirelessly send the robot these commands:
- UP: move up 1 square
- DOWN: move down 1 square
- LEFT: move left 1 square
- RIGHT: move right 1 square

The robot cannot pass through the walls shown as thick lines. If a command would cross a wall, the beaver remains in place waiting for the next command. Which of the following command sequences is the shortest that can help Natasha bring the robot to the square containing the star?

A computer image is stored in a square grid; each small square is called 1 pixel. A video is stored as a sequence of such images.



For example, the picture above shows a video recording a 10×10 square moving in a 20×20 frame. Each image needs 20×20=400 pixels of storage. The video has 11 images, so it needs 400×11=4400 pixels of storage. However, each image differs from the preceding one in only a few details. To save storage, only the first image and the changes in each next image relative to the preceding one are stored. How many pixels are needed to store this video?

Six cats are going to the mice’s party by jumping across islands in a river. The cats may only jump to adjacent islands. They cannot jump diagonally or jump onto or across submerged islands. Each cat stays on an island for exactly 1 minute before jumping to another empty island. Jumping time is negligible. If no neighbouring island is empty, the cat must remain for another minute.



What is the minimum number of minutes for all six cats to reach the mice’s party?