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I don’t believe it is possible to dissect a square into four Pythagorean triangles (a Pythagoren 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 a rectangle with sides 272 and 273 into four Pythagorean triangles?
How about a rectangle with sides 672 and 676?
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?
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?
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?
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?
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.
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?
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?
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?
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?
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:
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.
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?
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.
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?
How many ways are there of arranging the digits 1,2,3,4,5,6 such that the first two digits make a multiple of 3, the middle two digits make a multiple of 4 and the final two digits make a multiple of 5?
(Example: 123645, 12=3x4, 36=4x9, 45=5x9)
I walk 10 miles south, then 10 miles east, 10 miles north, then finally 10 miles west.
I end up more than 25 miles from where I started.
How is this possible?
I roll three standard dice, resulting in three random numbers between 1 and 6.
I bet you that you can’t make a number that is a multiple of ‘n’ by arranging the three numbers displayed on the dice to form a three-digit number.
‘n’ can be any ODD number larger than 1, which YOU nominate but you must do so BEFORE you see the results of the dice rolls.
For instance if you chose 13 and the dice rolls were 1, 6 and another 1, you could win by noting that 611 is a multiple of 13.
What is the best choice for ‘n’ to maximise your chances of winning?
Consider a list of all of the three digit numbers that:
are a multiple of 4
contain three different digits
contain at least one odd digit
don’t contain a zero.
Prove that for every number that satisfies these criteria there is exactly one other that uses the same three digits in a different order and also satisfies the criteria.
I have in mind a number, that is the smallest non-zero number that:
when doubled it is a square number,
when tripled it is a cube number,
when multiplied by 5 it is a power of 5, and
when multiplied by 7 it is a power of 7.
How many zeroes does it have after the last non-zero digit?
Although this puzzle might involve very large numbers, it can be approached without a computer or even a calculator.