How to solve the brain-teasers set by a top puzzle master
On Tuesday the Magazine published a story by maths writer Colm Mulcahy about the late puzzle master extraordinaire, Martin Gardner.
Prof Mulcahy took the opportunity to set readers 10 questions drawn from the vast number of puzzles Gardner wrote about in his 80-year career. We answered the first three questions straight away, and promised to provide answers to the others on Friday. So here they are.
For those who missed the questions first time round, we will repeat them here, along with the hints kindly provided by the professor, and - right at the bottom - the answers. (In the original story the questions were numbered 4-10, but here we will number them 1-7.)
1. An Englishman (Mr Salmon), a Welshman (Mr Green), and a Scotsman (Mr Brown) met for lunch one day. One man was wearing a salmon tie, another was wearing a green tie and the third was wearing a brown tie. "Isn't it funny," said Mr Brown to the others, "that not one of us is wearing a tie which matches our name?" "That's true," agreed the man wearing the green tie. Can you now say what colour tie each man was wearing?
2. Can you fold up a one-by-seven strip like this to form a cube with sides one unit long?
3. Can you think of two common words that begin and end with "he"? (No four-letter words please.)
4. We discussed on Tuesday how, when two squares of the same colour are removed from a chessboard, it is impossible to cover the remaining 62 squares with 31 dominoes each the size of two squares. But then we posed this question: suppose two squares of different colours are removed from such a board, for instance two adjacent corner squares. Show that the remaining 62 squares definitely can be covered with 31 dominoes each the size of two squares. This actually works no matter where the two squares are removed from. Can you construct a valid argument that works in all cases?
5. What is the significance of the repeated "little" in Lewis Carroll's All in a Golden Afternoon from Alice In Wonderland?
All in the golden afternoon / Full leisurely we glide; / For both our oars, with little skill, / By little arms are plied, / While little hands make vain pretence / Our wanderings to guide.
6. Can you fill in the blank space below to yield a true sentence?
In this sentence there are neither more nor less than ................... three-letter words.
7. Consider the magic square below. Note that its rows, columns and diagonals each add up to the magic constant 45. What else about it is interesting?
1. Who must have been wearing the green tie?
2. You're allowed to fold diagonally.
3. We know, it's painful until you get it.
4. No hint, sorry.
5. Who in real life inspired Alice In Wonderland?
6. Use your words. (Those of a mathematical bent should be able to suggest many solutions.)
7. When spelled out, 5 has four letters.
1. The green tie wasn't worn by Mr Green, we were told, nor by Mr Brown - since he just spoke to the man wearing it - and so, it must be worn by Mr Salmon. Then the brown tie must be worn by Mr Green, since neither of the other men could be wearing it, leaving Mr Brown sporting the salmon tie.
3. Headache and heartache (or a painfully easy solution could be "he").
4. In 1973, Ralph Gomory came up with the following ingenious argument. Consider a network of walls installed on the 8x8 board as shown, formed by two interlocked combs, one with three teeth, the other with four. In essence, this reconfigures the entire board as a 64-square maze: start on any square and move through the maze in either direction and you will eventually return to where you started, having visited every square exactly once.
Now imagine deleting any one white and any one black square from this path, as suggested in the image. This breaks the looped maze up into two snakes. Since the removed squares have opposite colours, each snake is composed of an even number of squares. A snake of even length can obviously be covered by dominoes, so it's now clear how to lay out 31 dominoes to cover the 62 remaining squares.
(This prompts even more questions, such as: "Can we still cover the remaining board with dominoes if we remove four squares, two of each colour?")
5. The word "little" in the Alice quote is Carroll's nod to Alice Liddell, the child who inspired him to write the tale. Jaap Engelsman in Amsterdam got in touch to tell us that Carroll used the word three times to reflect the fact that, as Gardner observed in his Annotated Alice book, on the Victorian day of the inspirational real life boat trip Carroll had with Alice Liddell, she was joined by her big sister Lorina and younger sister Edith.
6. "2" technically works, but bearing in mind the hint, "a pair of" is perhaps better. Other options could include "five minus three" or "nine minus seven" and so on.
7. When each number in the original square is replaced by the number of letters in its name - known as its logorhythm - we get a new square. There are four letters in "five", nine letters in "twenty-two", eight letters in "eighteen" and so on.
Amazingly, this too is magic, with magic constant 21. Even better, its entries are consecutive numbers. Subtracting 2 all around yields the classic lu shi magic square, with constant 15, known to the ancient Chinese.
British engineer Lee Sallows, who once wrote a research paper with Martin Gardner on another topic, discovered this magic linguistic/numeric tie-in, and wittily named the square starting with 5, 22 and 18 the li shu magic square.
But wait, there's more. Alert reader Tim from London quickly observed that the li shu magic square has another interesting property. Mulcahy checked, and its creator Lee Sallows doesn't recall noting this before.
"If you replace the contents of each cell with the sum of the digits in that cell, the square is still magic," Tim writes.
Vicki Powers of Arlington, Virginia, US, points out a simple explanation. In the case of the rows and columns, the same five digits are used each time, namely a 1, two 2s, one 5 and an 8, summing to 18 without fail. Admittedly, the digits on each diagonal sum to 18 for different reasons.
"Martin would have loved this, and I wish I could pick up the phone now and tell him," says Mulcahy. "His column thrived on reader response, and while the magic square puzzle here may be 'merely' entertaining, some of Martin's readers did discover new mathematics of real significance, such as the famous case of amateur mathematician Marjorie Rice."
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