«Contents Ü Foreword Elwyn Berlekamp and Tom Rodgers ½ I Personal Magic ¿ Martin Gardner: A “Documentary” Dana Richards ½¿ Ambrose, Gardner, ...»
In his October 1962 column, Martin Gardner presented a puzzle of mine that involved traveling through a city that had various arrows at the intersections. He used another of my puzzles in the November 1963 column — this one involved traveling in three dimensions through a ¢ ¢ grid. At the time I thought these were puzzles, but later I realized they were more like mazes. Around 1980 I started creating more of these things (which I now think could best be described as “mazes with rules”), and in 1990 I had a book of them published, Mad Mazes.
The next page shows one of the mazes from my book. This is my manuscript version of the maze, before my publisher added art work and dopey stories. (Actually, I wrote half the dopey stories and I sort of like some of them.) I chose this particular maze because it illustrates the cross-fertilization that Martin’s columns created. I got the original idea for this maze from remembering columns that Martin wrote in December 1963, November 1965, and March 1975. These columns presented rolling cube puzzles by Roland Sprague and John Harris. The puzzles involved tipping cubes from one square to another on a grid. As Martin’s columns said, you should think of a cube as a large carton that is too heavy to slide but that can be tipped over on an edge.
In my maze, place a die on the square marked START. Position the die so that the 2 is on top and the 6 is facing you (that is, the 6 faces the bottom edge of the page). What you have to do is tip the die off the starting square;
then ﬁnd a way to get it back onto that square. You can tip the die from one square to the next, and you can only tip it onto squares that contain letters.
The letters stand for ÐÓÛ,, Ó, and Ú Ò. If (and only if ) a 1, 2, or 3 is on top of the die, then you can tip it onto a square with an Ä. If a 4, 5, or 6 is on top, you can tip it onto a square with an À. If a 1, 3, or 5 is on top, you can tip the die onto a square with an Ç. If a 2, 4, or 6 is on top, you can tip the die onto a square with an.
I won’t give the solution, but it takes 66 moves.
Addendum, December 1998. Oops! My diagram is too big for this book.
The diagram should be at least 6 inches square to have a die roll across it.
You might try enlarging it on a copier, but you can also download it off my website. Go to ØØÔ »» ÓÑ º ØØºÒ Ø» ÖÓ Ø ÓØØ»ÖÓÐÐº ØÑÐ. While you’re there, check out the rest of the site. I have a long write-up (with pictures) of something called “walk-through mazes-with-rules." The ﬁrst of these walk-through mazes appeared at the Gathering for Gardner in January 1993. Since then the concept has grown. In the summer of 1998, several of the mazes were built as adjuncts to large cornﬁeld mazes.
Biblical Ladders Donald E. Knuth Charles Lutwidge Dodgson, aka Lewis Carroll, invented a popular pastime now called word ladders, in which one word metamorphoses into another by changing a letter at a time. We can go from ÌÀÁË to ÌÀ Ì in three such steps: ÌÀÁË, ÌÀÁÆ, ÌÀ Æ, ÌÀ Ì.
As an ordained deacon of the Church of England, Dodgson also was quite familiar with the Bible. So let’s play a game that combines both activities: Let’s construct word ladders in which all words are Biblical. More precisely, the words should all be present in the Bible that was used in Dodgson’s day, the King James translation.
Here, for example, is a six-step sequence that we might call Jacob’s Ladder, because ‘James’ is a form of ‘Jacob’:
Puzzle #1. Many people consider the Bible to be a story of transition from ÏÊ ÌÀ to ÁÌÀ. The following tableau shows, in fact, that there’s a Biblical word ladder corresponding to such a transition. But the tableau lists only the verse numbers, not the words; what are the missing words?
Puzzle #4. Find a Biblical word ladder from ÀÇÄ to ÏÊÁÌ.
Puzzle #5. (For worshippers of automobiles.) Construct a 12-step Biblical ladder from ÇÊ Ë (Judges 3 : 28) to ÊÇÄÄË (Ezra 6 : 1).
Puzzle #6. Of course #5 was too hard, unless you have special resources.
Here’s one that anybody can do, with only a Bible in hand. Complete the following Biblical ladder, which “comes back on itself” in an unexpected way.
Puzzle #7. Finally, a change of pace: Construct ¢ word squares, using only words from the King James Bible verses shown. (The words will read the same down as they do across.)
(Many other solutions are possible, but none are strictly increasing or decreasing.) Puzzle #4. Suitable intermediate words can be found, for example, in Revelation 3 : 11; Ruth 3 : 16; Ezra 7 : 24; Matthew 6 : 28; Job 40 : 17; Micah 1 : 8;
Psalm 145 : 15. (But ‘writ’ is not a Biblical word.) Puzzle #5. For example, use intermediate words found in Matthew 24 : 35;
Ezra 34 : 25; Acts 2 : 45; Ecclesiastes 12 : 11; Matthew 25 : 33; Daniel 3 : 21;
John 18 : 18; Genesis 32 : 15; 2 Corinthians 11 : 19; Psalm 84 : 6; Numbers 1 : 2.
(See also Genesis 27 : 44.)
References Martin Gardner, The Universe in a Handkerchief: Lewis Carroll’s Mathematical Recreations, Games, Puzzles, and Word Plays (New York: Copernicus, 1996), Chapter 6.
ØØÔ »» Ø ÜØºÚ Ö Ò º Ù» Úº ÖÓÛ× º ØÑÐ [online text of the King James Bible provided in searchable form by the Electronic Text Center of the University of Virginia].
Card Game Trivia Stewart Lamle 14th Century: Decks of one-sided Tarot playing cards ﬁrst appeared in Europe. They were soon banned by the Church. (Cards, like other forms of entertainment and gambling, competed with Holy services.) Card-playing spread like wildﬁre.
16th Century: The four suits were created to represent the ideal French national, uniﬁed (feudal) society as promoted by Joan of Arc: Nobility, Aristocracy, Peasants, the Church (Spades, Diamonds, Clubs, Hearts).
18th Century: Symmetric backs and fronts were designed to prevent cheating by signaling to other players.
19th Century: The Joker was devised by a Mississippi riverboat gambler to increase the odds of getting good Poker hands.
20th Century: After 600 years of playing with one-sided cards, two-sided playing cards and games were invented by Stewart Lamle. “Finally, you can play with a full deck!”—Zeus Ì Å, MaxxÌ Å, and BettoÌ Å, are all twosided card games.
Over 100 million decks of cards were sold in the United States last year!
“An odd number plus an odd number is an even number, and an even number plus an odd number is an odd number. OK?” “OK.” “An even number plus an even number is an even number. OK?” “Of course.” “An odd number times an odd number is an odd number, and an odd number times an even number is an even number. OK?” “Yes.” “Then an even number times an even number is an odd number. OK?” “No! It is an even number.” “No! It is an odd number! I can prove it!” How?
Move two matches so that no triangle remains.
Arrange the following ﬁve pieces to make the shape of a star.
Calculate the expansion of the following 26 terms.
´Ü µ´Ü µ´Ü µ ´Ü Ýµ´Ü Þµ
Which two numbers come at the end of this sequence?
2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, x, y
The ﬁgure shown here is the solution to the problem of dividing the ﬁgure into four identical shapes. Can you divide the ﬁgure into three identical shapes?
A 24-hour digital watch has many times that are palindromic. For example, 1:01:01, 2:41:42, 23:55:32, 3:59:53, 13:22:31, etc. (Ignore the colons.) These curious combinations occur 660 times a day.
(1) Find the closest such times.
(2) Find the two palindromes whose difference is closest to 12 hours.
(3) Find the longest time span without a palindromic time.
CREATIVE PUZZLE THINKING 39
A right triangle with sides 3, 4, and 5 matchsticks long is divided into two parts with equal area using 3 matchsticks.
Can you divide this triangle into two parts with equal area using only 2 matchsticks?
The legend of Sisyphus is a lesson in inevitability. No matter how Sisyphus tried, the small boulder he rolled up the hill would always come down at the last minute, pulled inexorably by gravity.
Like the legend, the physical universe has strange entities called black holes that pull everything toward them, never to escape. But did you know that we have comparable bodies in recreational mathematics?
At ﬁrst glance, these bodies may be even more difﬁcult to identify in the world of number play than their more famous brethren in physics. What, after all, could numbers such as 123, 153, 6174, 4, and 15 have in common with each other, as well as with various card tricks?
These are mathematical delights interesting in their own right, but much more so collectively because of the common theme linking them all. I call such individual instances mathemagical black holes.
The Sisyphus String: 123 Suppose we start with any natural number, regarded as a string, such as 9,288,759. Count the number of even digits, the number of odd digits, and the total number of digits. These are 3 (three evens), 4 (four odds), and 7 (seven is the total number of digits), respectively. So, use these digits to form the next string or number, 347.
Now repeat with 347, counting evens, odds, total number, to get 1, 2, 3, so write down 123. If we repeat with 123, we get 123 again. The number 123 with respect to this process and the universe of numbers is a mathemagical black hole. All numbers in this universe are drawn to 123 by this process, never to escape.
Based on articles appearing in (REC) Recreational and Educational Computing.
42 M. W. ECKER But will every number really be sent to 123? Try a really big number now, say 122333444455555666666777777788888888999999999 (or pick one of your own).
The numbers of evens, odds, and total are 20, 25, and 45, respectively. So, our next iterate is 202,545, the number obtained from 20, 25, 45. Iterating for 202,545 we ﬁnd 4, 2, and 6 for evens, odds, total, so we have 426 now.
One more iteration using 426 produces 303, and a ﬁnal iteration from 303 produces 123.
At this point, any further iteration is futile in trying to get away from the black hole of 123, since 123 yields 123 again. If you wish, you can test a lot more numbers more quickly with a computer program in BASIC or
other high-level programming language. Here’s a fairly generic one (Microsoft BASIC):
½ ÄË ¾ ÈÊÁÆÌ Ì ½¾¿ Å Ø Ñ ÐÐ ÀÓÐ » ´ µ ½ ¿¸ Öº Åº Ïº Ö
¿ ÈÊÁÆÌ ÈÊÁÆÌ Á³ÐÐ × ÝÓÙ ØÓ ÒÔÙØ ÔÓ× Ø Ú Û ÓÐ ÒÙÑ Ö ÒÓÛº
ÈÊÁÆÌ Á³ÐÐ ÓÙÒØ Ø ÒÙÑ Ö× Ó Ú Ò Ø×¸ Ó Ø×¸ Ò ØÓØ Ðº
ÈÊÁÆÌ ÖÓÑ Ø Ø Á³ÐÐ ÓÖÑ Ø Ò ÜØ ÒÙÑ Öº ËÙÖÔÖ × Ò ÐÝ¸ Û ÐÛ Ý×ÈÊÁÆÌ Û Ò ÙÔ Ö Ò Ø ÑØ Ñ ÐÐ ÓÐ Ó ½¾¿ººº ÈÊÁÆÌ ÇÊ Ä ½ ÌÇ ½¼¼¼ Æ Ì ½¼ ÁÆÈÍÌ Ï Ø × ÝÓÙÖ Ò Ø Ð Û ÓÐ ÒÙÑ Ö Æ° ÈÊÁÆÌ ¾¼ Á Î Ä´Æ°µ ½ ÇÊ Î Ä´Æ°µ ÁÆÌ´Î Ä´Æ°µµ ÌÀ Æ ½¼ ¿¼ ÇÊ Á ÁÌ ½ ÌÇ Ä Æ´Æ°µ ¼ ° ÅÁ °´Æ°¸ Á ÁÌ¸½µ ¼Á ° ÌÀ Æ ¼ ¼ Á Î Ä´ °µ»¾ ÁÆÌ´Î Ä´ °µ»¾µ ÌÀ Æ Î Æ Î Æ · ½ ÄË Ç Ç ·½ ¼ Æ Ì Á ÁÌ ¼ ÈÊÁÆÌ Î Æ¸ Ç ¸ ÌÇÌ Ä ¼ ÆÍ° ËÌÊ°´ Î Æµ · ËÌÊ°´Ç µ · ËÌÊ°´ Î Æ · Ç µ ½¼¼ ÈÊÁÆÌ Î Æ Ç ÎÆ·Ç ¹¹¹ Æ Û ÒÙÑ Ö × Î Ä´ÆÍ°µ ½½¼ ÈÊÁÆÌ Á Î Ä´ÆÍ°µ Î Ä´Æ°µ ÌÀ Æ ÈÊÁÆÌ ÓÒ º Æ ½¾¼ Æ° ÆÍ° Î Æ ¼ Ç ¼ ÇÌÇ ¿¼ If you wish, modify line 110 to allow the program to start again. Or revise the program to automate the testing for all natural numbers in some interval.
What Is a Mathemagical Black Hole?
1. Once you hit 123, you never get out, just as reaching a black hole of physics implies no escape.
2. Every element subject to the force of the black hole (the process applied to the chosen universe) is eventually pulled into it. In this case, sufﬁcient iteration of the process applied to any starting number must eventually result in reaching 123.
Of course, once drawn in per point 2, an element never escapes, as point 1 ensures.
A mathemagical black hole is, loosely, any number to which other elements (usually numbers) are drawn by some stated process. Though the number itself is the star of the show, the real trick is in ﬁnding interesting processes.
Formalized Deﬁnition. In mathematical terms, a black hole is a triple (b, U, f), where b is an element of a set U and f: U U is a function, all
1. f (b) = b.
2. For each x in U, there exists a natural number k satisfying f (x) = b.
Here, b plays the role of the black-hole element, and the superscript indicates k-fold (repeated) composition of functions.
For the Sisyphus String½, b = 123, U = natural numbers ¸ and f (number) = the number obtained by writing down the string counting # even digits of number, # odd digits, total # digits.
Why does this example work, and why do most mathemagical black holes occur? My argument is to show that large inputs have smaller outputs, thus reducing an inﬁnite universe to a manageable ﬁnite one. At that point, an argument by cases, or a computer check of the ﬁnitely many cases, sufﬁces.