I’d intended to publish this a few weeks ago but somehow forgot.
Why is Friday April 18th special, and why is Wednesday September 23rd equally special?
Puzzle of the Week #516 - Two Pairs of Circles
In a particular right-angled triangle, two circles tangent to the long leg have a radius or 5, and two circles tangent to the short leg have a radius of 4.
What is the radius of the incircle?
Puzzle of the Week #515 - Three Circles in a Trapezoid
Three circles are positioned inside the trapezoid below, such that they are tangent to the straight lines (but not tangent to the other circles). Given the two lengths confirmed in the diagram, what is the overall height of the trapezoid?
Puzzle of the Week #514 - Two More Circles in Trapezoids
In this shape, two circles are inscribed in two right trapezoids. The acute angles of the trapezoids add to 90 degrees. What is the overall height of the shape?
Puzzle of the Week #513 - Two Circles in a Trapezoid
Given this construction below, where two circles are tangent to the straight lines in the diagram, if we are told that A+C = B+D what is the value of A/B?
There are two possible answers.
Puzzle of the Week #512 - 4x4x4 Cube
You might have heard of the puzzle of slicing a 3cm x 3cm x 3cm cube into 27 1cm cubes, and how, even if you are allowed to rearrange the pieces between cuts, it still takes a minimum of six cuts to perform this action. There is a very clever argument that proves it.
Now consider a 4cm x 4cm x 4cm cube, cut into 64 1cm cubes. If you aren’t allowed to rearrange the pieces, will take nine cuts as shown below.
The question is: in the 4x4x4 case, with how few cuts can we slice it into 64 cm cubes if we ARE allowed to rearrange the pieces between cuts?
Puzzle of the Week #511 - Seven Circles
A triangle has an incircle of radius 5.
If two identical circles are placed within the triangle such that they are both tangent to the base, tangent to each other and each tangent to one of the other sides of the triangle, those circles have a radius of 4.
If seven identical circles are placed on the base of the triangle, all tangent to one another in a chain, and the first and last circles tangent to the other sides of the triangle, what is the radius of those circles?
As a bonus question, how many circles of radius 1 can you fit in a tangent chain along the baseline within the triangle?
Puzzle of the Week #510 - Pair of Circles 2
This is more difficult that the previous puzzle, but could benefit from insights learnt from solving that puzzle.
Given a triangle with sides 13, 14 and 15, and a pair of non-overlapping identical circles within the triangle, what is the maximum radius those circles could be?
Puzzle of the Week #509 - Pair of Circles
Given a triangle with sides 21, 28 and 35, and a pair of non-overlapping identical circles within the triangle, what is the maximum radius those circles could be?
Puzzle of the Week #508 - Jigsaw
Place the jigsaw pieces into the grid to make a valid crossword. The eight given pieces belong in the eight outer spaces in the grid. The central square is not given: you must reconstruct it yourself. This missing central piece comprises four letters and no blanks.
Puzzle of the Week #507 - Triangle in a Quarter Circle
In a quarter circle of radius 14, a right-angled triangle is placed such that two of the vertices are 10 units from the centre of the circle, and the third, a right angle, lies on the arc of the circle. What is the area of the triangle?
Puzzle of the Week #506 - Hexagon and Circle
A circle is tangent to the midpoint of one of the sides of a regular hexagon with side length 2x. A line going between diametrically opposite points as shown has length of 1 one side of the circle, and of x (half the side length of the hexagon) the other side of the circle.
What is the value of x?
For a bonus, what is the radius of the circle?
Puzzle of the Week #505 - Linked Values
In this isosceles triangle, values of ‘a’ and ‘b’ are chosen such that the sides of the triangle are ab, ab and ab/2, and that the line shown going from ‘a’ away from the left vertex to ‘b’ away from the right vertex forms a right angle. This isn’t enough information to define a and b, however if you know one of them it is possible to calculate the other.
What are all of the solutions where both a and b are integers?
Puzzle of the Week #504 - Sum and Product
If the sum of a and b is 100, what is their product?
Puzzle of the Week #503 - Semicircle in a Triangle
Here is a semicircle within a triangle. What is its radius?
Puzzle of the Week #502 - Almost a Square
* Edited with input from Graham Holmes and Philip Morris Jones.
I don’t believe it is possible to dissect a square into four Pythagorean triangles (a Pythagorean triangle is a right-angled triangle where all the sides are whole numbers). But there are some rectangles that are close to a square that can be so dissected.
Can you dissect each of these rectangles into four Pythagorean triangles?
168 x 169
252 x 253
272 x 273
One of these uses four identical triangles, one uses two pairs of identical triangles, and one uses four different triangles (although two of the triangles are similar).
Puzzle of the Week #501 - Rectangle and Quarter Circle
A quarter circle and a rectangle are tangent and/or coincident at three point as shown. What is the length and width of the rectangle?
Puzzle of the Week #500 - Rectangle
Two unit-radius circles are positioned in a rectangle such that they are each tangent to two edges and the diagonal of the rectangle.
What do you get if you subtract the value of the area of the rectangle from the value of the perimeter of the rectangle?
Puzzle of the Week #499 - Octagon to Square
This octagon with a flower shaped hole in the middle can be split into eight identical parts which, when half of them are flipped, can be reassembled to for this square with five circular holes in it.
What proportion of the square is shaded?
Puzzle of the Week #498 - Three Week Odyssey
I embark on a trek, walking a certain distance on the first day, and then each subsequent day walking a mile further than the previous day.
For the first week I head east. For the second week I head north. For the third week I head directly towards my starting point. At the end of the three weeks I am back where I began.
How far was my trek in total?