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 #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 #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 #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?

Puzzle of the Week #492 - Grid Fill 2

We saw last week how it was possible to fill in a 7x7 grid according to certain rules, but that there were only two solutions (ignoring reflections or rotations).

Despite being smaller, a 6x6 grid has at least five distinct solutions. Can you find them?

 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.

A solution cannot be simply a reflection or rotation of another solution.

Here is one of the solutions to the 7x7 grid, and five 6x6 grids to fill in:

Puzzle of the Week #491 - Grid Fill

This puzzle is loosely based on the Japanese puzzle Shakashaka.

It involves partially shading an n by n grid.

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). 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.

 I’ve shown an example with a 3x3 grid; the grid you need to solve is 7x7.

Puzzle of the Week #490 - Inheritance

Three brothers Adam, Barnaby and Charles are each granted a share of their father’s estate upon his death, such that the eldest brother Adam receives the most, the next eldest Barnaby a lesser amount, the third son Charles the least.

 

Charles comes up with a scheme such that there will be three rounds of redistribution.

Firstly Charles takes half of his share and splits it equally between the other two. Then Barnaby does the same, and finally Adam does likewise.

Miraculously they all end up with the exact same amount!

 

If instead, they did the exact same procedure but beginning with Adam, and proceeding through Barnaby, then Charles, Adam would end up with £19,881 more than Charles.

 

How much were they each granted in the will?

Puzzle of the Week #489 - Triple Coin Flipping

Coins are placed within a rectangular grid. Some are showing heads, others tails, at random.

You can select coins, three at a time (three adjacent coins in a line, horizontally, vertically or diagonally), and turn them over. Through a sequence of such moves, it is possible to make it so that all of the coins are showing heads.

Find the arrangement with the fewest coins, such that any starting arrangement of heads or tails can be made all heads by a series of three-in-a-row flips in horizontal, vertical or diagonal directions.

The coins must lie in a rectangular grid pattern. You can have spaces in the grid without coins, however a triplet of coins you wish to flip cannot bridge across any gaps.

Puzzle of the Week #488 - Tournament

There are 16 teams in the knockout stages of a football tournament. For the sake of argument, say that each team has a distinct ranking from 1 (best) to 16 (worst), and that in any individual match, the better team will always win and progress to the next round.

The first round matches are randomly arranged, and the eight winners of those matches are randomly arranged in four matches in the quarter final etc.

What is the probability that the fifth best team will reach the final?