Monthly Archives: December 2015

Christmas 2015 – a doublet

christmas treeCharles Lutwige Dodgson, alias Lewis Carroll, did not just write Alice in Wonderland, but also books on Mathematical subjects and puzzle books. The story goes that Queen Victoria, who reigned Britain during Charles live, was so enchanted by ‘Alice in Wonderland’, that she wrote the author and asked him to send her a copy of his next book. Charles dutifully did sent her his next book – on a mathematical subject.

Charles was also the inventor of a type of puzzle where one word has to be changed into another word by changing one letter at a time.
cot (replace ‘a’ with ‘o’)
cog (replace ‘t’ with ‘g’)
dog (replace ‘c’ with ‘d’)
Lewis Carroll says that he invented the game on Christmas day in 1877. The first mention of the game in Carroll’s diary was on March 12, 1878, which he originally called “Word-links”, and described as a two-player game. Carroll published a series of word ladder puzzles and solutions, which he called “Doublets”, in the magazine Vanity Fair, beginning with the March 29, 1879 issue. Later that year it was made into a book, published by Macmillan and Co.The one which Charles originally used was the problem to change HEAD into TAIL:
heal (Replace ‘d’ of ‘head’ to ‘l’)
teal (Replace ‘h’ of ‘heal’ to ‘t’)
tell (Replace ‘a’ of ‘teal’ to ‘l’)
tall (Replace ‘e’ of ‘tell’ to ‘l’)
TAIL (Replace ‘l’ of ‘tall’ to ‘i’)
The puzzles have been called Doublets, Word-links, Laddergrams, Word-golf, and Word-ladders.

At this time of the year, a Christmas puzzle seems appropriate. Over the past century, attention at Christmas seems to have shifted from Mary and her Baby to the christmas tree.

Try to change the word MARY into TREE in the fewest number of steps. Or, if you prefer that, you can change the word TREE back to MARY.

Marcel Danesi, Ph.D., on, believes that ‘solving them will give the verbal areas of the brain a veritable workout. The reason I believe this to be the case is that a solution entails knowledge of both word structure and semantics. The main semantic process involved is word association and, thus, recall, which is a powerful form of brain-activating thinking, at least as I read the relevant research. We are of course faced with the usual problem of trying to understand or explain how the research translates into benefits through puzzle-solving. The way I look at it is that puzzles such as the doublet can only be beneficial to overall brain health. As one’s semantic memory begins to wane through the aging process, giving the semantic parts of the brain a puzzle workout can only be advantageous

You can check your solution here

You are welcome to remark on the puzzle: its wording, style, level of difficulty. I love to read your solution times. Please do not spoil the fun for others by listing the solution. Solutions will be posted after one or more weeks.

Sources and further reading: The Levehsteind distance between two words is the number of operations that is needed to change one word into another by adding a letter, removing a letter or replacing a letter. The Damerau–Levenshtein distance is identical, but also allows the transpostion of two adjacent characters.
The distance between two words in a doublet as used by Dodgson is a special case of the Levenshtein distance: inseryion and deletion are not allowed, while all intermediate words must appear in a dictionairy. doublets as a complex network

Bongard problem (5)

The Russian scientist M.M. Bongard published a book in 1967 that contains 100 problems. Each problem consists of 12 small boxes: six boxes on the left and six on the right. Each of the six boxes on the left conform to a certain rule. Each and every box on the right contradicts this rule. Your task, of course, is to figure out the rule.

Bongard problem rule 6 exercise 2015-12-02

You can check your solutions here

You can find more Bongard problems at Harry Foundalis site, and I intend to publish more problems in the future.

A new puzzle is posted every friday. You are welcome to comment on the puzzles. Solutions are added at the bottom of a puzzle after one or more weeks.

The 4 cards (Cont’d)

The brainteaser of the 4 cards is a nice teaser, which made me wonder if it could be generalized. Indeed I found a couple of ways to vary upon this theme.

1) The 3 values
There are six cards in front of you. Each of them has a letter on one side and a number on the other side. Three of them have letters face up: A, B and C. The other three have numbers face up: 1, 2 and 3.
How many cards (and which) do you want to check if you want to know every card with ‘C’ on the front face has a ‘2’ on the reverse?
6 cards

You can check your solutions here

2) The three triangular blocks
Another way to vary on this subject is to have more than one backside. Consider the wooden blocks depicted in this figure. They have three sides (plus a top and a bottom). One side has a letter, one side a color and one side a number. Only one side is facing you. You can only rotate them clockwise. You are not allowed to get up and walk around them.
As you can see, each block now has two ‘backsides’, a leftback and a rightback. The letter is either A or B, the number either 1 or 2, and the colour either orange or purple.
3 blocks

As you can see there is an ‘A’, a ‘2’ and an ‘Orange’ facing you.

How many rotations do you have to make to ascertain if the rightback of all B is purple?

You can check your solutions here