Earlier today, I presented you with these challenges from the Mathigon puzzle advent calendar. Here they are once more, along with their solutions.
The knights are approaching.
![puzzle](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights.png)
Solution
![When you see the solution it’s obvious! A knight on a black square can only attack a white square.](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-1.png)
The round where numbers are missing.
![puzzle-2](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-2.png)
Solution
Every number is equal to the sum of the two numbers before it. For example, 1 + 3 = 4, 3 + 4 = 7, and so on. This pattern is similar to the Fibonacci sequence, where the same rule applies but it starts with 0, 1.
3. Square are you?
![puzzle-4](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-3.png)
Solution
![solution-4](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-4.png)
4. Hexa-going, hexa-going, hexagon.
![puzzle-5](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-5.png)
Solution 3/16
![solution-6](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-6.png)
5. Satsuma segment
![puzzle-3](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-7.png)
Solution 25/16
This problem involves more steps. You will need to apply Pythagoras’s theorem for right triangles as well: the square of the longest side is equal to the sum of the squares of the other two sides. To start, extend the semicircle to create a full circle and draw a triangle underneath it. The circle intersects with the left side of the square at its midpoint. The left angle of the triangle is a right angle because any point on a circle forms a right angle with the diameter.
![solution-3](https://assobeleyme.org/wp-content/uploads/2024/01/have-you-successfully-solved-it-mischievous-knights-8.png)
Thus d2 = a2 + (1/2)2 + 5/4
and d2 = (1 – a)2 + 1
Thus a2 + 6/4 = 1 – 2a + a2 + 1
2a = 2 – 6/4 = 1/2
= 3
The value of d equals 3 when a equals 1/4.2 = 25/16, so d = 5/4
I trust you had a great time solving today’s puzzles. I’ll return in 14 days…on December 25th!
Mathigon, owned by the global educational technology company Amplify, is a highly interactive and innovative online platform for mathematics. Best of all, it’s completely free! Be sure to give it a try.
Since 2015, I have been posting a puzzle every other Monday. I am constantly searching for new and interesting puzzles. If you have a suggestion, please email me.
Source: theguardian.com