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?

Puzzle of the Week #485 - Best Bet

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?

Puzzle of the Week #484 - Twin Numbers

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.

 

 

 

Puzzle of the Week #483 - Trailing Zeroes

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.

Puzzle of the Week #482 - Number Triangle 2

Similar to last week, but now you have to arrange the numbers 1-15 in a triangle such that each number is the difference of the two numbers immediately below it.

This time there is only one possible solution (plus its reflection), but even if you cleverly narrow down the possibilities with a couple of quick observations, there are still over 60000 triangles to check without a computer.

Alternatively, if I were to give you the bottom row, completing the grid will be trivially easy.

In an effort to find a sweet spot, I have given you the numbers on the left hand slope. Some more numbers can be placed straight away, whereas others you can only initially narrow it down to a couple of possibilities.

Puzzle of the Week #480 - Wordle Cities

Presumably you’re familiar with the online game of Wordle but here’s a brief recap: you try to guess a solution word. If your guess word has any correct letters in the correct place they are coloured green. If your guess has correct letters but not in the correct place these are coloured yellow. If the solution word only has one E for instance, but your guess has two Es and neither are in the correct place, only the first will be coloured yellow.

 With that covered, I have come up with a variation. I have selected three English cities, and treated each of them as both the solution word and the guess word in all possible combinations, and coloured them according to Wordle rules. So for instance cities 2 and 3 start with the same letter, and cities 1 and 3 share a middle letter.

Your task is to identify the cities.

Puzzle of the Week #477 - Twenty Sum Thing

Fill in all the empty squares of the grid with the digits 1-9, such that:

 1:            A diamond shape denotes that the four numbers adjacent to it add to 20.

2:            Every diamond uses a different way to add to 20, so if for example one diamond is surrounded by 1,2,8,9, no other diamonds can use those same four numbers, not even in a different order.

3:            Identical digits are not allowed to occupy adjacent squares of the grid, not even diagonally adjacent.

Puzzle of the Week #475 - Letter Pairs

Use the following letter pairs to form a series of words: a 4-letter word, a 6-letter word, an 8-letter word, a 10-letter word and a 12-letter word.

Each of the letter pairs can only be used once, EXCEPT for five of them.

Those five special letter pairs appear at the end of each of the five words, and also earlier in the same word. For instance if the 8-letter word ends in OR, it would need to be either:

(OR----OR), (--OR--OR) or (----OROR).

 

AH  AM  BA  ID  IO  ME  NA  NC  NT  OR  OU  RA  SE  SY  TA