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?

Puzzle of the Week #497 - Six Regions

Here is a 1512 x 1512 square, which is divided into six regions of equal area.

What are the coordinates of the marked point?

A clue to help you: all of the junction points in the diagram have integer coordinates.

Puzzle of the Week #496 - Four Towns

I wish to travel to Torton which is generally in the north-east direction of where I’m starting in Kipton.

If I go directly from Kipton to Torton it takes me 78 minutes.

Lawton is a town 15 miles due east of Kipton.

If I go from Kipton to Lawton and then on to Torton it takes me 102 minutes.

Gunton is a town 16 miles due north of Kipton.

If I go from Kipton to Gunton then on to Torton it takes me 82 minutes.

You can assume there are straight roads between each of the towns and I always travel at a constant rate.

How far is it from Kipton to Torton?

Puzzle of the Week #495 - Smallest Triangle

The triangle at the left of this figure is such that if the base is extended to the right, and a line is drawn through the apex such that it forms equal angles with the other two sides of the triangle, the length of the dashed line is 45.

There are infinitely many such triangles, however we will restrict ourselves to triangles with integer lengths. And we wish to find the smallest such triangle (either by area or by perimeter – it’s the same triangle).

Can you find it?

Puzzle of the Week #494 - Five Dice

You roll five standard D6 dice, obtaining a five digit number abcde.

abcde is not divisible by 2, nor is it divisible by 7.

Moreover, none of the possible rearrangements of abcde are divisible by 2 or 7.

The second roll (b) is greater than the fourth roll (d).

What was the exact sequence of dice rolls?

Puzzle of the Week #493 - Grid Fill 3

This is the third and final instalment in this series about Shakashaka-like grids.

We look now at an 8x8 grid. There are at least 19 distinct solutions, so I’m not going to ask you to find them all. Instead I want you to find a particular one. All but one of the solutions I have found have at least one axis of symmetry. The asymmetrical solution has 5 black squares.

 A reminder of the rules

You can diagonally half-shade some of the squares of the grid (shown below in grey).

You can fully shade some of the squares at the edge of the grid, but none in the interior (shown below in black) and if a square is fully shaded no adjacent squares can also be fully shaded.

The regions that remain unshaded must form only diagonal squares and diagonal rectangles.

 Here is one of the symmetrical solutions. Can you find the asymmetrical solution?