Category Archives: Mathematics

Alice and the sweets


This week we have an honoured guest – Alice. Yes, Alice from Alice in Wonderland!
Alice heard Tweedledum say: Yesterday we got a number of wine gums. We both got the same number, but we played a game which I won and then I had 5 times as many wine gums as Tweedledee. But when I gave him one of my wine gums, I had only 4 times as many wine gums as he had.

This week we have an honoured guest – Alice. Yes, Alice from Alice in Wonderland!
Alice heard Tweedledum say: Yesterday we got a number of wine gums. We both got the same number, but we played a game which I won and then I had 5 times as many wine gums as Tweedledee. But when I gave him one of my wine gums, I had only 4 times as many wine gums as he had.

1) How many wine gums did Tweedledee and Tweedledum have?

That’s strange, Alice said. Yesterday, I mean the day before I fell into this rabbit hole, I had been playing a game with my sister. We were playing for matches, and after the game I had 7 times as many matches as my sister. But when I gave her one of my matchsticks, I had only 6 times as many matches.

2) How many matches did Alice and her sister have?

I was telling the story above to a little girl called Alais. Alais thought for a moment, then told me: That is strange. Yesterday, the day before you told me this puzzle, I was playing a game with my little sister. After the game, I had 10 times as many gumdrops as she had. Of course I gave her one, and then I had exactly 9 times as many as she had.

3) Alais is obviously a very smart girl. I knew she had had some elementary algebra at school. She had obviously figured out that there are an infinite number of numbers with which this puzzle can be told. Can you explain why?

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are usually published after one or more weeks.

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Maze with numbers


This week I’d like to present a new type of maze.

Somewhere among the concept of this blog is a concept about mazes, their history, their shapes and their applications. In my small library are at least two good books about mazes. But here’s a new type of maze. Smaller versions may be suitable in the class room, while this shape and size are aimed at adults.

Your task? You see the two big 7’s? Find your way from one to the other, by moving vertically or horizontally, through squares which are a multiple of 7. Sounds easy, does it?

Here’s your exercise:

You can check your solution here

New puzzles are published at least twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to discuss difficulty levels, variations and alternate solutions, but plz. don’t publish the solutions.

Knights tours


Today we have “special forces”, such as the SAS in the UK, the Russian Spetsnaz, and the USA Rangers and Seals. In the middle ages these special forces had a name that still rings today: knights. The link between them and todays puzzle is very thin: the knight got a place in western chess, and todays puzzle uses the move of the knight on the chessboard.

In the series on new magazine format puzzles, I published a post on a new word format puzzle I encountered in the free newspaper metro in my native Netherlands.

In it, you have a 3×3 grid, the center of which is empty, while the outer edge is filled with letters. The letters form a word, and consecutive letters are always a knights jump (knight in chess) apart. Example:


For those who don’t know how a knight in chess moves: move one horizontally or vertically, followed by a diagonal move away from the starting square.

A knight on the square marked “K” may move to any square marked “X”.

My main criticism is that the puzzles as published by Metro are too easy to solve.

Today it occurred to me that the size of the board can be increased, and the size altered, to increase the difficulty of the puzzles.

9 letters**/*****


12 letters***/*****


20 letters****/*****


That increased size raises the difficulty level of the puzzle is easy to understand: a larger size does not only give more starting positions, but also more possible moves on subsequent moves.
Another way to increase the difficulty is by not using single letters – the human mind in the western world is used to work with them – but digrams (two letter combinations) or trigrams (three letter combinations).

Proverb split into digrams***/*****

Proverb split into trigrams***/*****

A new puzzle is published at least twice a month. I welcome your comments below, but please do not spoil the fun for next visitors by listing your solutions. Solutions are published one or more weeks later. You can find more puzzles with words by following the link to the right.

You can check your solutions here

Hidden numbers


In issue @@@ of @@@, Sanders published a new type of puzzle, called ‘hidden numbers’.

I must confess that the puzzle was too hard for me, though in the future I may give it a try again.

In this post I present a simplified version.
1. The numbers 1 to n have been hidden in a square grid.
2. Yellow areas give the sum of the hidden numbers in row and column of the yellow square.

Example:


1) 4×4*/*****


2) 5×5*/*****


3) 6×6**/*****


From here there are two ways to increase the difficulty of the puzzle (aside from increasing the size):
a) Put more than one hidden number in a row and / or column. This is what Sanders did.
b) Sum only the first number visible in any row or column. Any number, including the numbers in yellow squares, block the line of sight for any numbers behind them.

New puzzles are published twice a month on Fridays. Solutions are published after one or more weeks. You are welcome to discuss difficulty levels, variations and alternate solutions, but plz. don’t publish the solutions.

Round about


1) Nr 1**/*****

1 Uttered short, shrill sound
2 occurring more typically than an alternative form
3 place for keeping explosives
4 found in the earthcrust
5 give and receive reciprocally
6 make anew
7 follow along behind
8 live forever
9 more than opponents
10 leave country

The letters around each numbered square are an anagram of the numbered clue.
The numbered squares should be filled with the first letter of each word of the solution
Together, the letters form a word.

This puzzle is a variation on the “Blokje om” puzzle, in Visie 2017 nr 16. Visie (vision) is a magazine published by de Evangelische Omroep (Evangelical Brodcast) in The Netherlands.

You can check your solution here

New puzzles are published at least twice a month on Friday. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Nonogram


Sanders, the most innovative puzzle magazine publisher in my native Netherlands, recently published a new variation of Nonogram, also referred to as Hanjies, grid puzzles, picross or, in Dutch, Japanese Puzzles (Japanse puzzels).

It uses triangular grids instead of the customary square grids, which adds a nice touch to this puzzle type. This puzzle type is sometimes called Triddlers, and comes in two types:
(a) the triangles are the half of a square.
(b) These triangles are equilateral triangles, with 60 degrees in every corner.
The puzzles in this publication arte of the latter type, which I much prefer. A nice feature is the inclusion of a small puzzle alongside a big one on every page. Well recommended!
You may wish to try to obtain one through their webshop, I just discovered they now ship internationally.

(Apologies for not including a sample puzzle this post, I havent yet discovered good tools for making good triangular or hexagonal grids. Tips are appreciated, plz add them in the remarks section)

TooT


In this post I’d like to introduce TooTs, a mix between crossword puzzles and numbers. The grid looks just like a crossword puzzle, but instead of words the grid has to be filled with numbers. Vertical numbers must be read top-down. Thus if the digits 3, 9 and 5 are listed from the top down, the number would be 395.

Every clue consists of three numbers. Two of them have to be added together to get the number to be filled into the grid.
Example: the clue is 7, 8 and 13. Then the solution is either 7+8=15, 7+13=20 or 8+13=21. The name TooT is shorthand for Two out of Three.

Here is a 5×5 exercise:
Toot 5x5 2015-04-24 exercise

Horizontal
1) 16, 17, 18
3) 20, 26, 36
4) 142, 139, 145
8) 6819, 20002, 30134
11) 18, 20, 22
12) 11, 24, 36
Vertical
2) 17, 19, 23
3) 18, 36, 47
5) 400, 406, 418
6) 18, 106, 256
7) 15, 25, 190
9) 1, 51, 61
10) 11, 12, 13

A 7×7 exercise:
Toot 7x7 2015-04-24 exercise

Horizontal
1) 16891 18930
6) 382, 23, 67
8) 25, 8, 17
10) 32, 14, 17
11) 2913476, 173823, 1876543
12) 61, 23, 38
13) 45, 11, 34
14) 865, 249, 444
16) 13947, 1171, 5419
Vertical
2) 53, 26, 27
3) 8843269, 332160, 345612
4) 22, 3, 5
5) 12263, 5321, 6942
7) 62652, 23487, 39165
9) 591, 109, 482
10) 374, 25, 98
14) 83, 16, 26
15) 54, 17, 27

You can check your solution here and here

A 9×9 puzzle:
Toots 9x9 2015-05-15 nr 1

Horizontal
1. 108, 132, 146
4. 2, 166, 660
6. 2497, 9892, 12837
9. 0, 7, 24
11. 212, 669, 774
12. 4, 19, 30
13. 18, 27, 27
15. 14, 33, 40
16. 242, 977, 2236
17. 596, 903, 2770
18. 25, 31, 52
20. 4, 11, 22
21. 7, 9, 35
22. 126, 343, 422
24. 3, 10, 13
26. 2918, 74181, 82214
28. 292, 320, 398
29. 66, 191, 228
Vertical
1. 38, 96, 224
2. 4, 41, 77
3. 239, 1644, 4146
4. 19, 29, 35
5. 3, 7, 227
7. 20, 36, 38
8. 1, 14, 17
10. 12591, 13966, 31881
12. 706, 10961, 36955
14. 186, 210, 367
15. 102, 153, 279
19. 2287, 3330, 3945
21. 112, 239, 304
22. 19, 26, 45
23. 6, 23, 87
25. 74, 299, 315
26. 33, 49, 52
27. 12, 12, 12

You can check your solution here and here

In a subsequent post, probably next month, I hope to publish some variations.

Matchstick – triangles


Surface***/*****
These 6 matches enclose an area with size 2 (2 standard triangles).
Rearrange them so that:
a) they enclose an area of exactly 4 triangles
b) they enclose an area of exactly 6 triangles

matchstick-triangles-2-2016-12-14-nr-1-exercise

You can check your solutions here.

New puzzles are published at least twice a month on Friday. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.

Calcudocu / K-doku / Calcdoku


Calcdoku problems, also called K-doku or Calcudoku were invented in 2004 by Japanese math teacher Tetsuya Miyamoto, who intended the puzzles to be an instruction-free method of training the brain. He used the Japanese name KenKen, which could be translated as ‘Cleverness’. In his classes, he sets aside about 90 minutes each week for solving puzzles. He believe that when students are motivated, they learn better, and he lets them do so at their own pace.
Other names used for this type of puzzle are Kendoku and Kashikoku naru Puzzle. The names KenKen and Kenduko are trademarked. Books are in Japan published by Gakken Co. In the USA the New York Times started publishing them in 2008. In my native Netherlands they appear regularly in both puzzle magazines and general magazines.

A calcudoku puzzle consists of a latin square – a latin square can have any size. If its size 4, the numbers 1 to 4 should appear exactly once in every rown and column exactky once. Similarly, if its size 5, the numbers 1-5 should appears exactly once in every row and column.
The square is subdivided into smaller areas, and the sum, product, difference or division result is given in the top left corner.

Here are two example puzzles.
1) 4×4**/*****
calcudoku-4x4-2017-01-10-nr-1-exercise

You can check your solution here

2) 5×5**/*****
calcudoku-5x5-2017-01-12-nr-1-exercise

You can check your solution here

3) 6×6****/*****
is an example puzzle, kindly offered by Jacques Min, who runs a website specialized in Calcudoku puzzles: :
calcudoku_justpuzzles

Techniques for solving them:
* http://www.conceptispuzzles.com/index.aspx?uri=puzzle/calcudoku/techniques
*

You can check your solution here

New puzzles are published at least twice a month on Friday. Solutions are published after one or more weeks. You are welcome to remark on the difficulty level of the puzzles, discuss alternate solutions, and so on. Puzzles are rated on a scale of 1 to 5 stars.