Puzzles
Santa
Each of the letters D, A, Y, S, N, T, B, R and E represents a different non-zero digit. The following sum is true:
$$
\begin{array}{cccccc}
D&A&D&D&Y\\
B&E&A&R&D&+\\
\hline
S&A&N&T&A
\end{array}
$$
This has a unique solution, but I haven't found a way to find the solution without brute force. This less insightful sum is also true with the same values of the letters (and should allow you to find the values of the letters using logic alone):
$$
\begin{array}{ccccc}
R&A&T&S\\
N&E&R&D&+\\
\hline
S&A&N&E
\end{array}
$$Shooting hoops
Source: Alex Bolton
You spend an afternoon practising throwing a basketball through a hoop.
One hour into the afternoon, you have scored less than 75% of your shots. At the end of the afternoon, you have score more than 75% of your shots.
Is there a point in the afternoon when you had scored exactly 75% of your shots?
The self referential crossword
Source: Alex Bellos's Monday puzzle
Fill in the following crossnumber grid so that each clue describes the solution.
![](javascript:showlimage("self-ref.jpg"))
For example, if some clues read "TEN DS", "ONE X" and "THREE ES" then there will be ten Ds, one X and three Es in the completed grid. The entries in the crossword include the spaces.
![](javascript:showlimage("self-ref-ans.jpg"))
Cutting corners
Source: New Scientist Enigma 1773
The diagram below shows a triangle \(ABC\). The line \(CE\) is perpendicular to \(AB\) and the line \(AD\) is perpedicular to \(BC\).
![](javascript:showlimage("cut-corn.png"))
The side \(AC\) is 6.5cm long and the lines \(CE\) and \(AD\) are 5.6cm and 6.0cm respectively.
How long are the other two sides of the triangle?
Rugby scores
In a rugby (union) match, 3 point are scored for a kick, 5 for a try and 7 for a converted try. This scoring system means that some total scores can be achieved in different combinations, while others can be achieved in only one way.
For example, 14 can be scored in two ways (three kicks and a try; or two converted tries), while 8 can only be achieved in one way (try and a kick).
What is the highest score which can only be made in one way?
What is the highest score which can be made in two ways?
Quarter circle
Source: Maths Jam
A quarter circle is drawn in a square. A rectangle is drawn in the corner of the square which touches the circle and has sides of length 8 and 1.
![](javascript:showlimage("quarter-circle.jpg"))
What is the length of a side of the square?
![](javascript:showlimage("quarter-circle-ans.jpg"))
Shinro
Source: AMS Grad Blog
In the grid below, there are 12 holes hidden in the grid. Each of the arrows is pointing towards at least one hole. None of the arrows covers a hole.
![](javascript:showlimage("shinro.png"))
For example, the arrow in the first row tells you that at least one of the last four squares in that row contains a hole.
Find the location of all 12 holes.
![](javascript:showlimage("shinro-ans.png"))
Dice maze
Source: Daily Telegraph, 30/05/2015
Each colour represents a different direction (up, down, left or right) and the number of dots on each die tells you how far to go. Starting at the middle die, following the directions correctly will cause you to visit every die exactly once.
Which die will you visit last on the journey?