How many rhombuses?

In Mathematics, a Rhombus is a figure consisting of 4 lines, all of the same size, and with opposing sides parallel.
Thus:

How many rhombuses do you count in this figure?

You can check your solution at here

Please try to solve the puzzles on your own: your self confidence will grow. You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the soultions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

Slitherlink puzzle

Slitherlinks are one of 20+ puzzle types invented by or first published by the Japanese puzzle firm Nikoli.

The rules are simple: draw one line through the grid, in such a way that
a) it forms one closed loop, and
b) in each cell the number of borders matches the number in the cell, and
c) the loop may not touch itself, not even its corners.

Example:

 

 

So try your hand at a small puzzle:
1) 5×5

 

 

 

There are good guides with solving techniques at:
* nikoli
* English wikipedia
* Conceptis puzzles

2) 8×8

 

 

 

3) 9×9

 

 

 

4) 10×10

 

 

 

5) 5×5 honeycomb
We need not restrict ourselves to squares. Here is one example with a honeycomb.

The last one is a screenshot from the Loopy program, which you can freely download from Simon Tatham sportabel puzzle collection.

Dissections – the Greek cross (2)

A few weeks ago I posted some . Unfortunately I overlooked some of the puzzles created by Sam Loyd, so here is part 2. Sam Loyd must have been really fascinated by this shape, judging from the number of times he wrote about it.

1) Red Cross Volunteers

Here is a pretty little crutting puzzle, which is said to have originated in the mind of a red cross lassie while serving in Uncle Sam’s Ambulance Corps. It is safe to say that the bright witted little volunteer must have been a lineal descendent of Betsy Ross, who, it will be remembered designed the five-pointed star with one deft clip of her scissors. In the present instance it was necessary to practice strict economy in the manufacture of the red crosses to decorate the arms of the nurses, for the reason that the supply of red flanel was running very short in camp, so the problem presented is as follows: take a square pices of paper and without any waste cut it into five pieces which will fit together so as to make two Greek crosses of the same size.
This problem appeared on pages 199 of Sam Loyds cyclopedia of puzzles.

You can find the solution here

2) Divide the greek cross into three pieces

To illustrate the principle of working a puzzle backward, according to the axiom that a good rule should work both ways, we introduce a seasonable problem wherein the object is to discover how to divide a cross into three pieces which can be fitted together so as to form a rectangle which is twice as long as it is wide.
This, of course, is merely reversing the proposition of converting a rectangle into a greek cross, but, in that it presents the angles which must be fitted together, it is not so difficult as the other proposition.
This problem appeared on page 46 of Sam Loyds Cyclopedia of puzzles.

You can find the solution here

3) A swiss puzzle

Here is a very pretty trick performed by Miss Carré Schwitzer, which rivals Betsy Ross’ feat of producing a five pointed star with one clip of the scissors. When admiral Schwitzer asked his daughter to suggest an ensign for the Swiss navy, Carré seized an odd shaped remnant of red wall paper and skillfully divided it in two pieces which would fit together so as to form the Swiss flag with the white cross, as shown in her left hand.
When she was told of Betsy Ross’ feat she said she could go her one better. She took a Swiss flag, as here shown, and cut it into two pieces which would fit together and form a perfect square.

Of course if you can make a Swiss flag from a square, it is just as easy to reverse the operation – cut a square in two pieces which will form a flag.

4) The storm signal
Carré performed other feats with the Swiss flag which we will take occasion to mention. When she had charge of the signal station on Mt. Pilatus and whished to signal the fleet that a storm was rolling down the mountain, she took a square piece of bunting and cut it into two pices which would fit together and form the following flag.

In the Swiss language this tells of an approaching storm. Literally translated it says: “There will be a hot time in the old town tonight.”
Just to see how clever Miss Schwitzer was, try to cut the signal flag in two pieces which will form a perfect square.

5) Miss Schwitzer cont’d
Miss Schwitzer always acted on the square and was much respected on that account. She taught her Sunday School class how to cut three little squares into the fewest possible number of pieces so as to form one big square, and also the way to cut the three squares so as to form a Swiss cross. Try both of these puzzles.

6) Wilhelm Tell
William Tell asked her how to make a Maltese cross and she replied “Pull its tail”. She founded the order of the red cross.

There are two very beautiful puzzles connected with this cross, which are worth knowing: Cut the cross in two pieces which will form a rectangle, or cut it in three pieces which will make a perfect square.
We shall take early occasion to mention some of the marvelous feats performed by Carré Schwitzer in cutting Swiss cheeses, and juggling with pans of milk at her Swiss milk factory, near the chalk hills of Luzerne.

Problems 3-6 can be found on page 14 of Loyds cyclopedia of puzzles.

You can find the solutions for:
nr 3,
nr 4
nr 5
nr 6

7) The greek cross
There are also a number of puzzles on page 58, but I think they are largely overlappping with the puzzles above.

Divide the clock

The illustration shows an old fashioned analog clock.
Usings two straight lines, in how many parts can you divide it so that the digits on all parts have an identical sum?

I found this puzzle on aplusclick.com as a former math olympiad problem.

I would like to encourage you to solve this puzzle on your own. It will increase your self confidence, while looking up the answer will lower your self esteenm.

When you have solved this puzzle, you can check your solution here

You are welcome to remark on the puzzles, and I love it when you comment variations, state wether they are too easy or too difficult, or simply your solution times. Please do not state the solutions – it spoils the fun for others. I usually make the solution available after one or two weeks through a link, which allows readers to check the solution without the temptation to scroll down a few lines before having a go at it themselves.

Letterboggle

This puzzle type has nothing to do with the game “letter boggle”.

As usual, the rules are simple:
– fill in the entire alphabet, using every letter exactly once;
– consecutive letters are adjacent horizontally, vertically or diagonally;
– a letter in the border means that the letter appears in the indicated diagonal, row or column;

In the following puzzle, some letters are given to help you start:
1) Letterboggle^*

I found this type of puzzle in “PUZZEL”, a year-end addendum to the newspaper reformatorisch dagblad. It had 2 puzzles of this type, copyrighted M. Balmaekers, though Mr. Balmaekers used a 5×5 square and dismissed the letter Q. I intend to have more puzzles like this in one of the upcoming e-books.

I would like to encourage you to solve this puzzle on your own. It will increase your self confidence, while looking up the answer will lower your self esteenm.>When you have solved this puzzle, you can check your solution here

Magic snake

The magic snake is a plastic puzzle manufactured by Shuo Yi toys factory, Shang Hua town, China. It is constructed of a series of half cubes, cut diagonally, and connected with what loooks like a string. I don’t know the price, it’s a present given by “Black Pete”.

The packaging looks cheap, and the back carries the images of 9 3D figures which can be constructed with it.

Here are some more figures which you may wish to create:

Dissections – the Greek cross

The Greek cross consists of 5 squares joined in the shape of a cross.

1) Greek cross – 4 equal parts^*

The figure above shows a strangely formed meadow between a brook and mountains. There are 4 wells in the area. The farmer died and stipulated in his will that his land would be distrubuted equally among his 4 sons; all 4 lots would have the same area and shape and contain exactly 1 well.

How was the land divided among the 4 sons?

2) The hindu problem

The greek cross as shown in the illustration to the left, is composed of 5 equal sized squares. Cut the cross into five pieces to form a square. There are 2 solutions. According to British puzzle master Henry Dudeney, the problem is over 3000 years old.

3) How many straight cuts?
Henry Dudeney next comes up with the problem:
How many cuts do you need to divide the Greek cross into 4 pieces, so that with these pieces you can form a square?

4) The Red CRoss Lassie
American puzzle master Sam Loyd recounts the following problem:

In the whole realm of puzzledom, and geometry included, there is nothing so fascinating and eminanetly scientific as the series of problems pertaining to the form of the Greek cross and its peculiar relations to the square, parallelograms and other symmmetrical shapes.
As differing from the well known mathematical problem of converting the cross into a square by the fewest possible number of cuts, attention is called to the following pretty feat of chaninging one cross into two.
It appears that one of our wounded boys in blue who was returning home after being nursed back to life by a faithful Red Cross lassie, begged the red cross from her arm as a keepsake; but she, in true sweetheart style, took her scissors and by a few deft clips, cut the red cross into several pieces, which could be fitted together perfectly so as to make two crosses of similar vsize. It is a simple but beautiful trick, and the satisfaction of guessing it will be as great as if you should win a prize.

You can check your solution here for no 1, here for no 2 and here for no 3, and here for no 4

A new puzzle is published every friday. The solution is generally published one week later. I welcome your reactions on these puzzles: are they too easy, too difficult, are there any multiple solutions? How long did you need to solve it?

Parks

A Park puzzle has only two rules:
1) Every row, column and park has exactly 1 tree;
2) Trees are not adjacent horizontally, vertically or diagonally.

1) The stats^*

2) parks 7×7^*

You can check your solution here and
here.

Did you know?
My current customer has the nice habit of allowing its employees a certain amount of freedom. It aint as much as Google’s former 20% free time, but it does offer facilities such as posting reflective sayings. One I came across is:

If you think adventure is dangerous, try routine. It’s lethal. (Paulo Coelho)

This may be not be literally true of the body, but I believe it’s certainly true of the mind. Brains which do not regularly encounter new challenges, develop less well when young and detoriate faster when old.

One thing I consciously try to do is presenting new puzzles. And not just new puzzles, but also present a new type of puzzle. This means your brain has to start afresh with a new problem. You have to figure out new ways to tackle this challenge.
By presenting several puzzles of the same new type your brain has a chance not only to discover HOW to solve them, but also to let these ways reach the conscious state. You realize what the new tricks are with which you can solve these problems. And that is an important element of acquiring new skills (and I suspect for your brain an important part of creating new neural connections)

Pattern codes – circles

1) Circles^**

Which code belongs at the question mark?
If you wish, you can peek at a hint

I have long suspected that there is a strong connection between mathematics and puzzles. Proving such a relation according to the scientific standards is of course another matter. It was nice to read that a study by the University of Chicago found that puzzle play helps boost Learning Math-Related skills in children between ages 2 and 4.

Pentominosa

1) 5×12 nr 1

1) 5 x 12 rectangle^*
This 5×12 rectangle consists of the twelve different figures of 5 squares each. The original borders have been removed.
Each figure contains the letters A, B, C, D and E exectly once.
Can you find restore the borders between the twelve figures?

For those of you who are not familiar with them, here are the 12 possible pentominoes, or possible figures of 5 squares.

This type of puzzle was, as far as I know, first published in the British magazine Games and Puzzles, issue no 50, july 1976.
There is a type of puzzle where all bones from the double 6 set are laid down in a 7×8 square. Sometimes that type of puzzle is called dominosa. You can find a couple of them on my homepage, the domino plaza. Because the type of puzzle is very similar, I have christened this type of puzzle pentominosa.

2) 5 x 12 rectangle^*

3) 5 x 12 rectangle^**

You can check your solutions at here, here and here.

Please do not list the solution(s). In all other respects, I welcome discussion, listing alternative solutions, and I espoecially welcome your solution times, as that helps me to get an impression of the difficulty.