In the train I overheard calling a woman her friend: “I saw five blouses, but I had only money enough for four of them. I could have bought four of them for 65,80, or a combination of four of them for 61,80, or four others for 58,80, or another combination for 57,80, or still another combination for 54,80. But I was just 5 cents short of buying all of them. How much money did she have with her?
You can check your solutions here
One of our daughters obtained a nice sliding puzzle app on her ipad, and it features various levels, each level with several hundreds of puzzles.
The app is also available on android as “unblock me”, by Kiragames. It is also available in Googles play store for windows, see https://play.google.com/store/apps/details?id=com.kiragames.unblockmefree&hl=en.
I tried the app and the puzzles start out easy, but soon turn out to present decent challenges. Reason enough for a post on this type of puzzles. I’ll limit myself in this post to 2d sliding block puzzles. There are many 3d sliding block puzzles too: puzzle collectors may remember the many secret boxes puzzles, often beautifully crafted by woodworkers. But those are worth a different post.
There are many types of sliding block puzzles. Sam Loyds 15 puzzle is probably the most famous one. Many sliding block puzzles have been computerized, and the Sokoban puzzles are perhaps the first type of sliding block puzzle that exists only as a computer puzzle and not as a mechanical puzzle. But in this post I’d like to take a closer look at the type of sliding block puzzle that at the English wikipedia is called Klotski. I feel some doubt at this name: it may be derived from polish, as the article says, but when I view the history of the article I think it is more likely that Klotski is the name of a computer or video game instead of the name of this type of puzzle. I have never encountered the name Klotski for this type of puzzle in any puzzle book.
Sliding Piece Puzzles (by Edward Hordern, 1986, Oxford University Press, ISBN 0-19-853204-0) is said to be the definitive volume on this type of puzzle. It lists 270 of his sliding block puzzles, all neatly categorized and with the solution in the shortest number of moves. That brings us of course to the question: what is a move?
Hordern lists 4 possibilities:
a) a sliding block is moved as many ‘units’ in one or more directions
b) a sliding block is moved one ‘unit’ in 1 direction
c) a group of sliding blocks is moved one or more units in 1 direction.
d) a sliding block is moved one or more units in one direction.
Option a) is most common among puzzlers, mainly because it corresponds with a physical action: move a block as far as you want without lifting your finger from the piece.
Horderns also subdivides sliding block puzzles in four categories:
I. Sliding block pieces – pieces move independent of each other.
II. Warehouse/soko puzzles: one piece pushes others.
III. Railway shunting puzzles – one or two pieces push or pull all the others
IV. Puzzles with plungers or levers.
I already treated railway shunting puzzles, and category IV is a group of rather rare and complicated puzzles. This post deals entirely with group I, and with the subgroup where pieces are rectangular and of unequal size.
The nice thing about mechanical puzzles is that you can patent them, such as USA patent 207,124.
I’m not sure about the early history of sliding block puzzles. Sam Loyds puzzles goes back to the 1870’s. One Henry Walton filed U.S. Patent 516,035 on 1893-03-14 for a sliding puzzle resembling 15-puzzle. According to Edward Hordern, this is the first even known sliding puzzle with rectangular blocks.
Horderns book “sliding block puzzles”, mentioned above, is the standard work of reference for this type of puzzle.
Links:
* http://www.novelgames.com/gametips/?id=121: Only 3 levels, but nrs 2 and 3 are a nice challenge
* http://www.spel.nl/game/sliding-block-puzzle.html – several classics
* http://www.cleverwood.com/about_sliding_block_puzzles.htm – general information and some links
* http://www.johnrausch.com/slidingblockpuzzles/
* http://home.comcast.net/~stegmann/sliding.htm – robs puzle page, examples of sliding block puzzles
Sam and Moshe start to explore a cave. They both have a torch and both torches start with the same length. Sam’s torch will burn 3 hours while Moshe’s torch will burn 4 hours. When they get out, they find that one torch has exactly three times as many centimeters left as the other.
How long have they been in the cave?
You can check your solutions here
(This puzzle was based on a puzzle found here)
Which one of the numbers does not belong in the following series?
3, 4, 8, 9, 10, 18, 19, 38
You can check your solutions here
“How many plant species are there?” It was of course my old friend professor Brainstrain who asked this. His nephew looked bewildered.
“10? No, more”, he thought. Then he asked: “over a hundred, maybe?”
“Many more,” the professor asked with a smile.
He went on:
“In the following addition, every digit had been replaced with a letter. Find the original sum.
You can check your solutions here
Which one of the numbers does not belong in the following series?
E, D, I, H, J, O, N, U
This kind of questions is one of several types found in the Wechsler IQ test. Unfortunately, one of the questions as I found it online has two answers.
You can check your solutions here
Apologies for not publishing a puzzle last Friday.
If there is 1 in 1, if there are 2 in 2, 3 in 7 but 0 in 8, how many are there in eleven?
You can check your solutions here
Cryptarithms, alphametics, verbalarithmetic are some of the names of a type of puzzle, where two, three or more words are given, and each letter must be replaced by a single digit. The most well known of this is:
1) Dudeneys classic**
SEND MORE -----+ MONEY
Replace each letter with exactly 1 digit and make it a correct addition. The example above is from Henry Dudeney.
(Solution: 10)
Verbal arithmetic puzzles are quite old and their inventor is not known. An example in The American Agriculturist[2] of 1864 makes the popular notion that it was invented by Sam Loyd unlikely. The name crypt-arithmetic was coined by puzzlist Minos (pseudonym of Simon Vatriquant) in the May 1931 issue of Sphinx, a Belgian magazine of recreational mathematics. In the 1955, J. A. H. Hunter introduced the word “alphametic” to designate cryptarithms, such as Dudeney’s, whose letters form meaningful words or phrases.
There are several types of cryptarithms. One of them is them is the double true. This type of cryptarithm shows an addition that is true in words, but can also be deciphered as a cryparithm. This puzzle comes from N. Tamura, who wrote a program to search for puzzles with a unique solution.
2) DOUBLE TRUE:**
THREE FIVE FIVE SEVEN TEN ------+ THIRTY
(Solution: 30)
3) Math formulas* are another subcategory.
This one was first published in Sphinx magazine, and republished by Jorge A C B Soares. He mentions M. Van Esbroeck as the author, and january 1933 as the date of first publication.
A B C = C4 B C A = D4
(Solution: 40)
Cryptarithms are of course language dependent. However, they are not limited to the English language. The booklet CIJFERWERK (digit work), written by J. van der Horst, published by Born periodieken, date unknown, isbn 8 710838 100910, has some 200 dutch cryptarithms. In Germany they are called “Kryptogramm”, in French “Cryptarithme”, or d’alphamétique. I see articles on them in the Japanese wikipedia, where they give the following example:
4) Japanese*
大宮 ×大宮 大井町 横浜 -----+ 浜松町
Please forgive me that my Japanese is insufficient to discover the author.
(Solution: 70)
5) List**
Lists are another subcategory of this puzzle type. Lists are composed of a number of items in category, followed by the name of the category, which equals to the sum of the items. In a good list items are unique, taht is, they appear only once.
Here is a non unique list, that is a list with items that appear more than once:
APPLE PEAR DATE APPLE PEAR DATE APPLE PEAR DATE APPLE PEAR DATE -----+ FRUIT
(Solution: 50)
Links
What is the code for the fifth figure?
This weeks puzzle is pretty easy, I admit. I hope next weeks puzzle will be more of a poser.
You can check your solution.
Shikaku puzzles are puzzles which can be found in some magazines. They were invented by Nikoli, a Japanese puzzle firm. Allthough they can be drawn in black and white, the colored versions seem to be more popular. There are several websites offering them – see below They are also known as Shikaku ni Kire, rectangles, Divide by Squares and Divide by Box.
The basic is a square or rectangle which has been subdivided into rectangles. The border lines are not shown in the exercise – this is what the solver has to find out. The sizes of the rectangles are given as clues.
Example:
The solution:
As you can see in the examples above:
(1) Only rectangles are used;
(2) Every rectangle has exactly 1 square indicating its size;
Here are some puzzles with them:
1) Problem 6×6
2) problem 7×7
3) problem 12×12
There are several apps for your android smartphone or ipad around. Sites which offer shikaku puzzles are:
justpuzzles 