Corey the beaver is having a party this weekend. He wants to meet many new friends, so he thinks of inviting people in rounds as follows:
- Round 1: Corey invites 4 guests.
- Round 2: Each invited guest invites 4 more people.
- Round 3: Each newly invited guest invites 4 more people.
- …
Assuming nobody is invited twice, what is the minimum number of rounds needed for the number of guests attending to be 1300?
Aika likes taking photographs of her cat and posting them on social media.
The dates on the original photographs are 24 March 2021, 14 June 2021 and 28 August 2021.
To manage them easily, she will name each photograph so that the computer stores the photographs in order from oldest to newest. Here are some naming methods Aika considers (the Vietnamese month-name tokens are preserved because their alphabetical order is part of the task):
- MEO_Thangtam_19_2021
- MEO_19_8_2021
- MEO_2021_Thangtam_19
- MEO_2021_08_19
- MEO_19_Thangtam_2021
- MEO_19_08_2021
- MEO_2021_19_8
- MEO_2021_8_19
The computer stores photographs by name in order from A to Z and from 0 to 9. How many of these naming methods can Aika use?
When using a computer camera, we often do not notice that our own image is reflected as if in a mirror. Four friends are having a group discussion. Each wears a shirt bearing their name.
A beaver uses a special method to encode messages. He uses the following table to turn letters into numbers. A message's value is the sum of the values of its letters.
| Letter | A | B | C | D | E | F | G | H | I | J |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| Value | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 |
For example, to encode “HI” he may write “72.73”. The value of “HI” is 145. The beaver has written these messages:
A turtle lives in a small garden. The garden is divided into square plots (white) and surrounded by a fence. The turtle can move between plots sharing a side, but cannot enter the stone squares. It wants to visit every plot in the garden exactly once. How many of the following gardens make this impossible?
Alex the beaver grows wheat in some parts of his field; the remaining parts contain only weeds. The following picture shows the pipe system that was installed earlier. Alex wants to block some pipes (at the junctions) so that all wheat-growing areas receive water and the remaining areas do not.
What is the smallest number of pipes he must block?
Stephan has seven stacks containing a total of 8 white-covered books and 8 gray-covered books, as shown below.
He wants to rearrange them into two identical stacks. In each move, Stephan may place any entire stack on top of another stack to create a taller stack. Which picture below cannot be his final result?
Five people go cycling together in the starting order shown below.
They normally ride at the same speed to preserve their order and keep relatively small gaps. However, every 15 minutes the person at the front slows down, allows the others to overtake, and moves to the last position.
The first person to finish completes a journey lasting 8 hours 39 minutes. Who is it?
Alex and Julia play a leaf-taking game with three plates containing 1, 2 and 3 leaves as shown below. On each turn, a player takes any positive number of leaves, but only from one plate. The winner takes the last leaf or leaves from the plates.
Alex plays first. From which plate should he take leaves on his first turn to guarantee a win, even if Julia plays as well as possible?
A park is divided into cells as shown below, and trees have been planted in some cells.
A monkey named Jack lives in the park. He can jump between trees exactly 2 cells apart in any direction.
Every day Jack climbs some tree and jumps between trees without going down to the ground. What is the largest number of different trees Jack can be on in one day?
A shop has four service counters numbered 1, 2, 3 and 4. Initially only counter 1 is open.
Each counter can hold at most 4 people, including the person being served if any. Service takes 2 minutes per person. If a customer arrives when the counters already have 4 people each, the next counter opens. However, a new counter needs 1 minute of preparation, so its first customer takes 3 minutes in total to finish; later customers take only 2 minutes.
A customer entering the shop checks counters 1–2–3–4 to find the first one with room and joins its queue. A customer arriving at exactly the time the front customer finishes service can still join that queue.
Initially two customers enter, and every minute two more enter, until there are 12 customers in total. After how many minutes will all 12 have been served?
Betty and Bobby exchange numbers using a special secret method. They encode digits according to the following table. Here ○ is a white circle and ● is a gray circle.
