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For the rest of us who teach teenagers, who often like to dabble in, or experiment with, things occultic or satanic, how can we make use of some symbolism of Halloween, such as black cats, snakes, and broomsticks, to teach some creative problem solving in mathematics? How can we bewitch or bedevil them with "dark math," by entertaining them with some "wicked thoughts" about the most disliked subject in school?
Below are a baker's dozen of dark math problems we could pose to kids, who could get an intellectual kick out of solving them. Who says math and fear can't positively mix? Didn't someone say that God created the natural numbers, and all the rest is the work of man—and of the devil?
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1. Dracula's Pets
Count Dracula collected an even number of insects (beetles and spiders) in his coffin. He counted 54 legs in all. How many spiders did he have?
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2. Friday the 13th and Halloween
Look at the following pattern:
13 × 13 = 169 and 31 × 31 = 961
Is this a mere palindromic oddity?
3. The Fewest Number of Coffins in a Day
On which day of the year do fewest people die?
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4. Cemetery Math
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R.I.P
Jasper Rice
Died: 31/10/85
Buried: 30/10/85
If there was no error or crime involved, explain this apparently impossible situation, which took place in October 1985.
5. The Secret Lives of Vampire Numbers
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Numbers such as 1435 are called vampire numbers (or just vampires) because they are formed when two parent numbers (also called fangs), 35 and 41, are multiplied together (35 × 41 = 1435). Another vampire number is 2197, because it contains digits of its fangs, 27 and 81 (27 × 81 = 2187).
(a) Are there any two-digit vampire numbers?
(b) How many 4-digit vampires are there?
6. A Deadly Will
A dead man's will states that seven bats were to be divided among his three ungrateful sons. The eldest was to receive one half of the bats, the middle one, one-fourth, and the youngest, one-eighth.
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They sought the help of a wise priest who suggested
that he would lend them one bat so that a “fair” allocation could be possible.
The eldest son would then receive 1/2 × 8 = 4 bats, the middle one, 1/4 × 8 = 2
bats, and the youngest one, 1/8 × 8 = 1 bat. Thus, the sons would receive 4 + 2
+ 1 = 7 bats, returning the borrowed bat to the holy man. Is the priest's solution
“fair”?
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In a complete period of 400 years (after this, the calendar repeats another similar cycle), which day of the week is Halloween mostly likely to fall?
8. Legion's Number
In the Gospel of Mark, Jesus encounters a multitude of demons possessing a man. When the Son of Man asked them, "What is your name?" the man replies, "My name is Legion, for we are many."
The computer scientist, Clifford A. Pickover, defined the number 666⁶⁶⁶ as Legion's number of the first kind, and the number 666!⁶⁶⁶! as Legion's number of the second kind.
(a) What are the last 31 digits of 666!?
(b) What are the last ten digits of 666!⁶⁶⁶!?
(c) Using a computer, what are the last ten digits of 666⁶⁶⁶?
By definition, 666! = 1 × 2 × 3 ×⋯× 665 × 666
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9. The Math of 9/11
Using the pattern and period of the calendar, and the day of the week September 11 falls this year, what day of the week was 9/11—the "Holy" 2001 War by the "fidels" on the Twin Towers?
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An equation uniting the Number of the Beast 666 and the golden ratio ø is given by
ø = −{sin 666º + cos[(6)(6)(6)º]}
Verify phi's apparent liaison with the satanic number.
11. A Grave Calculation
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How long will it take before the whole
world is covered in gravestones?
Hint: What is the total land area on Earth?
How much area does each person's grave take up?
The
Chinese avoid the "death number" 444; the Westerners avoid the "satanic number" 666. The West like the number 7 and multiples of 7, while the East like even
numbers but avoid odd numbers. In general, Asians' "lucky" numbers are
Westerners' "unlucky" numbers.
Why
do both the East and the West still have faith in numerology and astrology, in
spite of literacy, numeracy, and technology?
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It is said that Satan is a wolf in sheep's clothing: f(Santa) = Satan.
Prove that Oct. 31 = Dec. 25.
References
Pickover, C. A. (2002). The mathematics of Oz. Cambridge University Press.
Pickover, C. A. (1997). The loom of God: Mathematical tapestries at the edge of time. New York: London Plenum Trade.
Santos, A. (2009). How many licks? Philadelphia: Running Press.
Yan, K. C. (2011). CHRISTmaths: A creative problem solving math book. Singapore: MathPlus Publishing.
Selected Answers
1. 3 spiders and 5 beetles.
Spiders have 8 legs and beetles 6.
We need to solve 8s + 6b = 54 with s, b positive and s + b even.
Hence there are 3 spiders and 5 beetles.
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3. On the same day that the fewest people are born—29 February.
4. The man dies on a Monday and his body was then flown across the International Date Line to a place where it was still Sunday for immediate burial.
5. (a) No 2-digit vampires exist.
(b) 7 four-digit vampires.
8. (a) They are all zeros.
The last digits of 666! are 0 because 510 and 210 both divide 666! The number 666! has well over 100 zeros.
(c) The last 10 digits of 666⁶⁶⁶ are 0,880,598,016.
Challenge: What are the first 10 digits of Legion's number of the second kind?
9. Tuesday.
11. More than a million years.
Hint: How many people die every year round the globe?
Time taken = (Total area of the Earth) ÷ [(Number of dead people per year) × (Area per dead person)]
13. If "Oct." is the short form of "octal" and "Dec." for "decimal," then 31 (in base-8 notation) is equal to 25 (in base-10 notation): 31₈ = 25₁₀
© Yan Kow Cheong, October 31, 2012
