Bebras Colage B

Magic Drink Machine

Beaver Cobi has a mysterious blue coloured machine as shown in Fig. 1. There are two funnels in the machine. • If a beaver pours chocolate milk into both funnels, white milk comes out. • If a beaver pours white milk into either of the funnels, chocolate milk comes out. • If a beaver connects two machines and pours chocolate milk into both funnels as shown in Fig. 2, chocolate milk comes out at the end. Note: the middle green connection has no effect on the type of milk.

Question

What type of milk should be poured into both funnels so that white milk comes out when connecting three machines?

• chocolate milk - white milk

• white milk - white milk

• white milk - chocolate milk

• chocolate milk- chocolate milk

Damaged Secret Table

The secret writing here is based on encoding the letters of the Latin alphabet with new symbols. The symbols are described in the following table. Unfortunately parts of the table have been wiped out and some parts are missing: Question

Find the original plain-text of this cipher-text, even though the table is damaged.

• 
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Creating Numbers

Olivia the beaver, is playing with blocks. Each block has a single digit on it. She loves to make a big tower and then use the blocks one by one, from the top, to form a number. Each time she takes a block off the tower, she can place it to the right or to the left of the number she is forming. The following figures show a tower of 4 blocks and two possible numbers that can be formed from it (4235 and 2534): Olivia just built a new tower of 6 blocks and she wants to create the smallest possible number from it.

Question

Can you help her by dragging the blocks into their correct position?

• 653475

• 347565

• 346575

• 657534

Do you know how to add two letters? It’s easy: both letters become the numbers of their order in alphabet. The result will be a number, which we can turn into a letter by looking at which letter is in that position in the alphabet. For example, A+A=B, A+B=C, C+E=H If the resulting number is bigger than the number of letters in the alphabet, we can start again from the beginning of the alphabet, e.g. Z+A=A, Y+C=B Lenka uses this addition of letters to encrypt her journal. She works like this: 1. First she thinks of a number, which she calls shift. 2. She writes one word and then writes the same word again below the first one, but the second word is shifted to the right by a number of positions equal to the shift. For example, if the shift is 1, the encryption looks like this:

Question

Lenka encrypted the word COMPUTER, and came up with the word COMSJGUMTER. Which number is the shift?

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Epidemic Crisis

In Beaverland, there are 12 towns connected by highways as shown on the map below. Towns that are

connected directly or indirectly by one or more roads form an ‘economic community’. Currently, all 12

towns belong to the same community. Unfortunately, due to an epidemic outbreak, the mayors have decided, in order to reduce travel between towns, to close two highways (using roadblocks). Their goal is to split the country into three separate economic communities. As they want to minimize economic disruption, once the roadblocks

are in place, the smallest of the three resulting economic communities should contain as many towns

as possible. Question Choose the two roads they should close.

• G - I

• F - I

• A - I

• A - K

Aggelos the Mailman

A town on a lake has 6 houses where beavers live together or separately.

Aggelos is the new postal worker who doesn’t know anything about the beavers who live in the town.

Aggelos does not live in any of the 6 houses on the lake. At the beginning, Aggelos’ notebook is empty. He comes up with the following strategy to deliver the mail:

1. Every time a new letter is sent, Aggelos writes down the name and the address of the sender.

2. If the name of the recipient is in his notebook, he delivers the letter.

3. If the recipient’s name is not in his notebook, he makes copies of the letter and delivers them to every house in the lake except to the house of the sender. 4. In any case, the correct recipient always replies to the letter the same day. Aggelos then writes down their name and address and delivers the reply letter.

Question

On the first day Elia sends a letter to George and Mike sends a letter to Elia. On the second day Socrates sends a letter to Nasia.

How many letters has Aggelos delivered in his first two days of work?

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Math Machine

The beavers created a MathMachine. It takes a number as input and returns another number as output. Inside the MathMachine, there are many components. All components work in the same way. Each

component takes three numbers as input and processes them as follows: If the first number is 1, return the third number as the output of the MathMachine. If not: • Decrease the first number by 1. The result is the new first number. • Increase the second number by 2. The result is the new second number. • Add the new second number and the third number. The result is the new third number.

• Pass the new numbers to the next component, in the same order.

The first component is special: • When the MathMachine receives an input, it passes this number as the first input to the first component. • The other two inputs for this component are 1 and 1.

As soon as the MathMachine receives an output from any of its components, it returns this number as a result. Example The image shows how the MathMachine processes the input 2. This example uses only two of the MathMachine’s many components. Question

The MathMachine processes the input 4. Which number does the MathMachine return as output after passing through all four components?

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Wood Processing

A delivery of 100 wooden logs has arrived. All the logs are longer than 1m.

A team of four robots process the logs. Their task is to produce a full set of swing seats, as shown in

the images below. All four robots are able to select logs that still need to be processed and then perform their individual function on the selected log. The robots can work at the same time as each other if there are logs they

still need to process: Cutter: Cuts a log on one side and makes it 1m long.

Driller: Drills two holes through the log exactly 20cm from the left and 20cm from the right.

Printer: Prints the company’s logo in the middle of a smooth and clean log surface.

Remover: Removes the bark from logs of any length and makes the surface smooth and clean.

• A robot processes each log only once. • The robots can work at the same time as each other but not on the same log. • A robot starts working when it gets a start signal from the Control Program.

• From the moment a robot is started, it must never be idle, waiting for logs from other robots.

• A robot stops working once it has processed all logs, or if there are no more logs to process.

The commands in the Control Program are limited to those given below and are started sequentially.

Question Rearrange the commands in the Control Program so that the robots process all the logs without any of

them ever being idle.

• Start the Cutter -

  wait until all working robots have finished their work - Start the remover - Start the driller - Start the Printer

• Start the Cutter - Start the remover -

  wait until all working robots have finished their work - Start the driller - Start the Printer

• Start the Cutter - Start the remover -

  Start the driller - wait until all working robots have finished their work - Start the Printer

• Start the Cutter - Start the remover -

  wait until all working robots have finished their work - Start the Printer - Start the driller

Towers of Blocks

Sam the beaver is playing with his toy blocks. Each block is the same height, but has a different width.

He built nine beautiful towers, each one made with blocks of the same width. There are two ways to change the height of a tower: adding blocks to the top or removing blocks from the top. Either way, the energy cost to change the height of a tower is proportional to the width and the number of blocks being changed. Example

Removing 2 blocks from a tower with a width of 1 costs 2*1 = 2 units of energy; and adding 4 blocks to

a tower with a width of 3 costs 4*3 = 12 units of energy. Sam wants all towers to be the same height, and he wants to spend as little energy in total as possible. Question In total, what is the minimum amount of energy that it will cost Sam to make all towers the same height?

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Electric Car Queues

You have an electric car that travels at 100 km per hour and can travel 200 km when

fully charged. The car can be recharged at any time before the charge runs out. Each

recharge restores the car to full charge and takes 1 hour, regardless of how much

charge is left when it starts recharging. The picture below shows the roads

connecting A, B and C. You start at point A and want to visit points B and C in as little

time as possible. You can take any route — via B to C, or via C to B. When you pass

through point B or point C, you must stop for 30 minutes, but your car does not lose charge.

Along the route there are charging stations, each with a certain number of charging points. At each charging point, there may be cars currently in queue, and you have to wait for them to finish charging before you can use this charging point. For instance, at the charging station CP-AB1 there are three charging points, two of which are

empty and one which has a single car in queue. Each car waiting in queue takes 1 hour to charge.

The first car in each queue has just started charging when you set out from point A. You start with your car fully charged. Question What is the least amount of time it will take to visit points B and C, starting at point A?

• 5 Hours

• 6 Hours

• 7 Hours

• 8 Hours