Bebras 2022 - Level 4

There are three islands (the dark-gray regions) in the sea as shown below.



Two birds are on island C and want to reach island A. We know that:
- The birds can move only along the edges of the cells, up/down/left/right, and always choose the fastest route.
- The black bird can fly over 2 sea cells per hour and needs 1 hour of rest on land after flying across 4 cells.
- The white bird can fly over 4 sea cells per hour and needs 2 hours of rest on land after flying across 4 cells.
- On land, both birds walk at 1 cell per hour.

If they start at the same time, which bird reaches island A sooner?

Alice, Bean, Char and Dean occupy positions 1, 2, 3 and 4 in the picture, not necessarily in that order.



All four look at trees A, B, C and D and share their observations:
- Alice: “From left to right I see D–A–B–C.”
- Bean: “From left to right I see C–D–A–B.”
- Char: “From left to right I see C–A–D–B.”
- Dean: “From left to right I see D–C–A–B.”

Where is each friend standing?

A group of friends give one another gifts according to these rules:
- Nobody may give a gift to themselves.
- Every friend must give one gift.
- Every friend must receive one gift.

They record the actions with arrows pointing from giver to recipient. For example, the picture below shows cat giving to mouse and mouse giving to cat.



Which gift-giving arrangement below does not follow the rules?

A village has a wire system for transmitting information between houses, as shown below. Two houses (marked by red circles in the source) are disconnected and receive no signal from any other house.



Some wire sections must be repaired immediately to ensure every house receives a signal from at least one other house. The remaining damaged sections may be repaired the following week.

What is the largest number of wire sections whose repair can wait until the following week?

A witch wants to obtain a treasure by crossing the maze shown below.



Not knowing which treasure to choose, she decides to move according to these rules:
- Prefer moving down (↓).
- If moving down is impossible, prefer moving right (→).
- If there is no way down or right, move left (←).
- If none of those three choices is possible, fly up (↑).

Which treasure will the witch obtain by following these instructions?

A country has six cities connected by the roads below. Each number is the time in hours required to travel along that road.



Currently there is only one hospital, in city F. A second hospital is to be built so that everyone can reach a hospital as quickly as possible.

In which city should the next hospital be built?

Some fish swim in the following order.



When someone says two numbers A and B, with A<B:
- Every fish before position A (to the left in the picture) swims away. At the same time every fish after position B swims away.
- The remaining fish keep their positions and are renumbered from 1, front to back (left to right).

For example, if someone says 2 and 16, the remaining fish are:



Starting with the original 20 fish, perform these changes:
- Someone says 4 and 18.
- Then someone says 6 and 12.
- Finally someone says 2 and 5.

What is the resulting order of the remaining fish?

Eleven candidates have been chosen for an organization's executive committee. Some pairs are considered close to one another, through family, friendship, work and so on. In the following picture, line segments represent these close relationships: two people are close when they are joined by a segment.



To ensure impartiality, the committee must not contain two people who are close to one another.

What is the largest possible number of committee members?

A shop uses computer vision to recognize customers' actions. It can tell whether a customer is looking straight ahead, lowering their head or raising their head, using the bridge-of-the-nose length called nose.
- If nose=1, the customer is looking straight ahead.



- When the customer nods and lowers their head, the bridge of the nose appears longer, so nose>1.



- When the customer raises their head, nose<1.



To buy n tickets, the customer nods n times, then raises their head. The controller starts when a customer stands in front of the machine looking straight ahead.

Which comparison blocks should be inserted, from top to bottom, into the following code so the program behaves as required?



Original program transcription:
1. Set so_ve=0.
2. Repeat until ???:
- Wait until ???.
- Wait until ???.
- Increase so_ve by 1.
3. Sell so_ve tickets.

Comparison choices: (1) nose>0; (2) nose>0.8; (3) nose>1.2; (4) nose<0.8; (5) nose<1.1.

Anna and Bob place seven pawns in a row as shown below.



They take turns removing one pawn or two adjacent pawns. Whoever takes the last pawn wins. Both play very well, and Anna starts. Which of the following first moves should Anna make to guarantee a win? Dashed outlines show removed pawns.

Beavers often use beads to make decorations. To communicate easily they use symbols to describe their shapes: S for a star, T for a triangle, R for a rectangle and L for a line segment. They use three rules to shorten a description:
- Consecutive identical beads can be shortened by putting their number in front. For example, 4S means four consecutive star beads.
- A repeating group of beads can be described by its repeat count followed by the group's description in parentheses.
- Otherwise write the letter representing the bead.

For example, the bead string below can be written S 3(TR) 3S L, which contains 9 characters excluding spaces.



What is the fewest characters needed to describe the following string?

A beaver makes three-layer cakes: first shaved ice, then flavoring, and finally fruit. For example, the picture shows dark-gray shaved ice, chocolate shavings as the flavoring and two cherries as the fruit layer.



The shaved-ice layers must follow this order:

Red → Green → White → Blue

The flavoring layers follow:

Chocolate shavings → Breadcrumbs → Cookie crumbs

The fruit layers follow:

Cherry → Kiwi → Strawberry → Orange → Blueberry

The first cake has red shaved ice, chocolate shavings and cherries. What is the number of the next cake identical to the first?

To keep Bonnie from understanding Bean and Tom's conversation, they encode messages using 12 digits, each 0 or 1. Decoding has two steps:
- Step 1: Replace adjacent pairs of digits with A, B, C or D according to this table.

| Pair | 00 | 01 | 10 | 11 |
|---|---|---|---|---|
| Letter | A | B | C | D |



- Step 2: In the resulting letter sequence, replace consecutive pairs of letters according to this table.

| Pair | AA | AB | AC | AD | BA | BB | BC | BD | CA | CB | CC | CD | DA | DB | DC | DD |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Output | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |



After receiving an encoded message, Bean decodes it as C13.



Which code did Bean receive?

A Martian code has two parts:
- The first is the value of the word.
- The second encodes the order of the letters occurring in the word.

The table gives the letter values.

| Letter | A | B | M | N | O | R | S | T | U |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| Value | 1 | 2 | 4 | 10 | 50 | 180 | 300 | 650 | 960 |



For example, MARS is encoded as follows:
- Add the letter values: $4+1+180+300=485$.
- Sort the letters alphabetically: A–M–R–S. A is represented by 1; M, R and S are represented by 2, 3 and 4 respectively.
- Combine the results: MARS is encoded as 485–2134.

Which word below is represented by 2101–415632?

To prevent a wolf from sneaking into the garden and taking sheep, the beaver places dogs to guard the gates. One dog guards a region as shown below.



The wolf cannot sneak through gate 1, but can enter through gate 2. In the following garden, what is the smallest number of dogs needed to guard the gates?