The beavers have a common rule: always pay using as few coins as possible. For example, to pay 17 coins, they take one 16-value coin and one 1-value coin rather than seventeen 1-value coins. What is the smallest number of coins needed to pay 13?
Six beavers, NICOLE, KEVIN, KRISTINA, EDGAR, KATEY and NATHANAEL, play a game with their teacher. The teacher must find a beaver whose name satisfies these conditions:
- It contains G or I.
- Its last letter differs from the first letter of every other beaver’s name.
- Its first letter is also the first letter of another beaver’s name.
Paul the beaver has four stamps A, B, C and D, as shown.
He used stamp B four times to make figure 1. Paul used stamp B once and stamp D twice to make figure 2. To make figure 3, which stamp can Paul definitely not use?
The beaver chef has a safe for secret recipes. It is locked with the dial below.
To open the safe, the beaver turns the arrow alternately left and right to point to the letters in the password. For example, to reach B and then H, the beaver turns 1 step clockwise and then 2 steps anticlockwise, as shown.
How should the beaver turn the dial to open the safe if the password is CHEFDG?
The archaeologist Cleveria discovers the special sequence of symbols below.
After studying it, Cleveria finds how to decode the symbols using the following table.
Editable transcription of the table (shape names describe the original symbols):
| Outer shape / Inner mark | 1 stroke | 2 strokes | 3 strokes | 4 strokes | Circle | Circle + 1 stroke | Circle + 2 strokes | Circle + 3 strokes | Circle + top stem |
|---|---|---|---|---|---|---|---|---|---|
| Crown-shaped top | A | B | C | D | E | F | G | H | I |
| Pointed bottom | J | K | L | M | N | O | P | Q | R |
| Rounded bottom | S | T | U | V | W | X | Y | Z | — |
The table shows that the crown symbol enclosing a circle with three strokes represents H. Which sequence below is the decoding of the original symbols?
A black-and-white computer image is divided into a grid of small squares called pixels. Each cell is either white or black. Each row is then encoded by successive runs of white and black cells. For example, the letter “a” is encoded below. Each dash denotes a new row.
Koko the beaver keeps some animals in six enclosures A, B, C, D, E and F as shown below.
However, some animals can eat others, as shown above. For example, the wolf eats the sheep, while the worm may be eaten by the chicken. To keep the animals safe, Koko must put them in enclosures so that no animal is next to an enclosure containing an animal it can eat. Which arrangement below is unsuitable?
Eight beavers sit around a round table. We know that:
- Alice sits opposite David.
- Henry sits between Greta and Eugene.
- Franny sits next to neither Alice nor David.
- One beaver sits between Greta and Clare.
- Eugene sits to David’s left.
- The remaining beaver is Bruce.
Three beaver chemists have very large quantities of four substances (triangle, right triangle, filled square and square frame) and are trying to create a new four-point substance. The original substance symbols are shown below.
Each chemist will work for two days. However, only one chemist works each day, and only one chemical reaction can be performed in a day. A chemical reaction transforms several substances into a new substance. Each chemist knows the following reactions:
Editable transcription (the descriptive names refer to the original shapes):
| Chemist | First reaction | Second reaction |
|---|---|---|
| Carly | Triangle + right triangle → circle missing a quadrant | Circle missing a quadrant + arrow → circle |
| Paula | Square frame + right triangle → pentagon | Square frame + filled square → arrow |
| Barry | Filled square + pentagon → diamond | Triangle + circle + diamond → four-point product |
Which of the following work orders does not allow the chemists to complete their task?
The beavers put 13 balls in a triangular box as illustrated 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 immediately below it.
- There is at least one ball immediately below it that is in a dangerous position.
How many balls in the box are not in a dangerous position?
Editorial clarification approved for this practice package: the original wording above is preserved. For grading, interpret the two danger conditions with “at least one” (OR), as used in the official solution, rather than “at least two” (AND). The bottom edge of the box supports its bottom row.
The wizard Bibrax places 49 coins on a table. Bibrax closes his eyes and asks a beaver to flip any one coin. When he opens his eyes, he will tell the beaver which coin was flipped. The secret is: “The number of star-side-up coins in each row and each column was initially even.”
Editable transcription of the displayed coins:
| Row | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| 1 | Blank | Star | Blank | Star | Blank | Star | Star |
| 2 | Star | Blank | Blank | Blank | Star | Star | Star |
| 3 | Star | Blank | Star | Star | Blank | Star | Blank |
| 4 | Star | Blank | Blank | Blank | Blank | Star | Blank |
| 5 | Blank | Star | Blank | Star | Star | Star | Blank |
| 6 | Star | Blank | Star | Star | Blank | Star | Star |
| 7 | Blank | Blank | Blank | Star | Blank | Blank | Star |
In a machine, horizontal bars can tilt to the left (value 0) or to the right (value 1). When a ball falls onto a bar, it reverses that bar’s value and then falls as shown below.
Below are the changing bar states after two balls have fallen.
The sequences 0000, 0001, … represent the states of the bars at each moment. If three more balls fall one at a time, what will the state of the bars be?
