Puzzle of the Week #521 - Arithmetic Sequence

S is the sum of an arithmetic sequence with n terms:

 x_1 + x_2 + x_3 + … x_n-1 + x_n = S

 A subsequence that includes only every other term, but still starts at x_1 and finishes at x_n totals 88:

 x_1 + x_3 + x_5 + … x_n-2 + x_n = 88

 A subsequence that includes only every third term, but still starts at x_1 and finishes at x_n totals 60:

 x_1 + x_4 + x_7 + … x_n-3 + x_n = 60

 What is the sum of the whole sequence S?

 For bonus points if the first term is 0, what is the common difference between consecutive terms? 

Puzzle of the Week #520 - Irregular Octagon

I have a circle of radius 19, which sits inside of an octagon, which has equal sides but not equal angles. Each of the sides of the octagon has length 17, and each of the sides is tangent to the circle.

What is the area of the octagon?

Puzzle of the Week #519 - Word Pairs

In each couplet, the second answer is the same as the first except for the addition of one letter. If you collect the five extra letters you will spell a word.

A metal red or man in blue,
A blade that whirs or cuts in two.

A spring in stride, a triple move,
A dream you have, which time must prove.

Make it round, add pounds to excess,
Or squash it down with a heavy press.

Launch with lips, a sharp eject,
Or break apart and disconnect.

A scheme to mask premeditation,
Shake to wake from meditation.

Puzzle of the Week #518 - Four Rectangles

Four rectangles are arranged as in the figure, where at every circled point the corner of one rectangle coincides with the side of another. What is the height of the magenta rectangle?

Puzzle of the Week #517 - Friday April 18th

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).