Puzzles
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?
Integer part
Let \(\lfloor x\rfloor \) denote the integer part of \(x\) (eg. \(\lfloor 7.8\rfloor =7\)).
When are the following true:
a) \(\lfloor x+1\rfloor = \lfloor x\rfloor + 1\)
b) \(\lfloor nx\rfloor = n\lfloor x\rfloor\) (where \(n\) is an integer)
c) \(\lfloor x+y\rfloor = \lfloor x\rfloor +\lfloor y\rfloor \)
d) \(\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor \)
Make the sums
Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums reading across and down are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, \(4+3\times2\) is 14, not 10.
| + | - | = 4 | |||
| + | - | × | |||
| - | × | = 27 | |||
| - | × | ÷ | |||
| × | ÷ | = 16 | |||
| = 2 | = 8 | = 6 |
Rotating round table
At a large dinner, 24 people are to sit evenly spaced around a round table. Place cards are laid to show where everyone should sit. Unfortunately nobody notices the name cards and the guests sit down with nobody in the correct seat.
Show that it is possible to rotate the table so that at least two people will be in the correct seats.
Balanced sets
A set of points in the plane is called 'balanced' if for any two points \(A\) and \(B\) in the set, there is another point \(C\) in the set such that \(AC=BC\) (here \(AC\) is the distance between \(A\) and \(C\)).
For all \(n\geq3\), find a balanced set of \(n\) points.
![](javascript:showlimage("balanced-a1.png"))
![](javascript:showlimage("balanced-a1andahalf.png"))
![](javascript:showlimage("balanced-a2.png"))
![](javascript:showlimage("balanced-a3.png"))
121
Find a number base other than 10 in which 121 is a perfect square.