25 Christmaths Toughies from Singapore

Here are 25 non-routine fertile questions for friends and relatives to solve during this festive season:

1. The Recurring 25

Convert 0.252525... to a fraction.

2. The Sides of a Polygon

If the sum of the interior angles of a polygon is 2525°, what is the maximum number of sides the polygon can have?

3. December 25, 2025

On what day of the week does Christmas Day fall in the year 2025?

4. Number of Diagonals in a 25-gon

What is the number of diagonals in a 25-sided convex polygon?

5. How Many Ways?

In how ways can you arrange the letters of the word CHRISTMASTIME?

6. Merry Xmas & Happy New Year

If the value of MERRY CHRISTMAS is 189, the value of A HAPPY NEW YEAR is _____.

7. Santa's Routes

Determine the total number of distinct shortest routes from point A to point B in the following diagram grid map.

A

 

B

8. A Sum of 25 Powers

Find the sum of S = 2 + 2¹ + 2² + 2³ +⋯+ 2²⁵.

9. Writing 25

What is another way of getting the number 25 using the numbers 2 and 5 only once, other than equating it to 5²?

10. Time Needed

Determine how long it will take to return all the gifts mentioned in the song “The twelve days of Christmas” if the gifts are returned at the rate of one gift per day.

11. Join the Xmases Family

A mother who was born on December 25, married to a husband also born on December 25, gives birth to their first baby on December 25. What are the odds of the mother doing that?

12. Four Fours to Make 25

Using four fours, the four basic operations and, if necessary, √4 and/or 4!, form the number 25. [Note: 4! = 1 × 2 × 3 × 4]

13. The Largest 25-digit Integer

What is the largest 25-digit number that can be divided by 2 and 5 without any remainder?

14. Number of Zeros in 1 × 2 × 3 ×⋯× 24 × 25

How many zeros are there at the end of the product 1 × 2 × 3 ×⋯× 24 × 25?

15. The Chocolate Bar

Given a 25-piece square chocolate bar, how many snaps are required to break the bar into its individual pieces?

16. A Creative Thinking Question

It is late on Christmas Eve and little Noel is waiting the Christmas tree to be finished. At exactly what time will this happen?

17. Christmas and Halloween

Prove that Oct. 31 = Dec. 25.

18. Divisibility by 25

Show that, if n is a positive integer, 16n + 10n – 1 is divisible by 25.

19. A Pair of Christmas Alphametics

In each case, replace each letter with a digit, different letters being different digits.

(a) SANTA - CLAUS = XMAS

(b) A + MERRY + XMAS = TURKEY

20. A Sum of Squares

Find the value of 1² – 2² + 3² – 4² +⋯+ 25².

21. Number of Rectangles

A board has 25 squares. How many individual rectangles are there in all?

22. A Christmas Party

At a Christmas party, each child brought a present. Presents were put in a large basket. All presents were different but identically wrapped. Going home, each child randomly selected a present from the basket. What is the expected number of children who carry home their own presents?

23. How Many Lines?

Using only horizontal and vertical lines, what is the most number of straight lines you can draw between the dots in a 5 × 5 grid without lifting your pencil?

24. A Table of 25 Boys and 25 Girls

A total of 25 boys and 25 girls sit around a table. Show that it is always possible to find a person both of whose neighbors are girls.

25. All n's End in Zeros

For what values of n does 1 × 2 × 3 ×⋯× (n – 1) × n end in 25 zeros?

A Merry CHRISTmaths to all!

Selected answers
1. 25/99     2. 16 sides.         3. Monday.      4. 275 diagonals.     5. 778,377,600 ways.
6. 158.       7. 252 routes.      8. 2^26 - 2      9. 5 ÷ .2 = 25          10. 364 days.
11. 364/1         12. 4! + (√4 + √4)/4             13. 999…99990     14. 6.         15. 624 snaps.
16. Midnight.       18. 600.       19. (b) 2 + 97,445 + 6,928          20. 325.         
21. 225 rectangles.                    22. 1.                 23. 34 lines.      25. 105, 106, 107, 108, 109

For full solutions, consult the title in the Reference.

Reference
Yan, K. C. (2011). CHRISTmaths: A Creative Problem Solving Math Book. Singapore: MathPlus Publishing.

© Yan Kow Cheong, Dec. 16, 2011.

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LIFE and DEATH by Numbers

LIFE and DEATH by Numbers

One of Singapore’s founding fathers, Dr. Goh Keng Swee who was Mr. Lee Kuan Yew’s Economics tutor, and later his deputy prime minister, died at the age of 91 on May 14. And one of the best-known popularizers of recreational mathematics, and debunker of paranormal claims, Mr. Martin Gardner died at the age of 95 on May 22. Both men were blessed with a lifespan of more than four scores and ten, and hundreds of thousands of people benefited from those two selfless men’s contributions.

Some two decades ago, I bought a copy of Gardner’s book from an ex-colleague, and since then I had never looked back to enjoying the beauty and power of mathematics. Like thousands of Martin Gardner’s fans around the world, formal schooling often fails to expose us to the joy and beauty of mathematics, so when I first read Gardner’s Mathematical Carnival, it was numerical love at first sight.

Let's popularize and evangelize math

Many of us love math in school, because we simply like solving non-routine or challenging word problems, or we enjoy being challenged by logic games and puzzles, but few things come close to getting a glimpse of the beauty and utility of advanced math, as popularized by Martin Gardner for number addicts of all stripes.

Death is blind

Dr. Goh and Mr. Gardner were men of numbers who contributed much, and many of us reaped the fruits of their labors. As we ponder on their numerous outputs, it’s never too early for us to reflect on our own mortality, and what our 'mathematical legacy' would be when we too reach the end of life’s journey.

You’re just a heartbeat away!

You’d even die before you finish reading this post. I’d die while you’re reading it. Now! At this very moment! It’s easy to be preoccupied with our to-do (or not-to-do) lists, focusing on our teaching, writing, reviewing, or preparing for our next meeting. We go on our normal lives as if we've thousands of days or millions of hours left before our last breath. We don’t want to think that we’re all one heartbeat away, regardless of our position or station in life.

We operate on automatic mode: Today it’s just a normal day to do whatever we feel like doing. We forget that we’re a mist that appears for a while and then vanishes. We don’t want to admit that today we may die. Or, maybe we feel invincible that nothing mortal is going to affect us. We live as though our lives would go on forever. We’re like the foolish rich man who thinks like “Today or tomorrow I’ll travel or fly to that city or country, spend a few weeks there, carry on my business and make money,” not realizing that his life will be taken away the very night.

If we live up to three or four scores and ten, it’s hard to accept that by then no one will care what positions we held, how many titles we collected, how many books or articles we wrote, or which conferences we attended.

Live and work selfishly or selflessly 

People die while living and working selfishly. At your funeral, will colleagues and social media friends attend it to stretch the truth in order to make you look good, or to sound as if you’re not that mean, after all? Nobody would dare say an unkind word at that last gathering; there is an obligation or pressure to come up with something positive to say about the person who died. But we secretly think the same thing: He or she really wasn’t that great of a person or boss or politician.

Plan your eternal destiny now

Unless we realize that we’d be the next person in our family or company to die, we wouldn’t see why we needed to treat our colleagues well. We’ve to believe it enough that it changes how to live. To say NO to office politicking and scheming, to start treating our colleagues whom we don’t like decently and fairly, to accept others in spite of their idiosyncrasies, to recognize that we’re not as smart as we think we’re, or that we’re not above-average as we want others to see us. 

Think of your obituary and eulogy

Unless we’ve a holy fear of God who holds the key to life and death, we’ll foolishly live our professional or working lives as if we can earn our eternity by merely living decently; yet our hearts and minds are far from Him. We try to do or act good to get His attention, but the condition of our hearts remains deceitful and evil. May the Holy Spirit reveal the sinfulness of our minds and hearts!

Some earthly thoughts on the heavenlies and Hades

Let me close with some life-and-death thoughts, dressed in some quasi-numerical language.

 

Life is a canvas of if’s.

You are what or how you count!

Life & Death

The probability of one’s living up to 444 years old is 0.

The probability of one’s dying is 1.

Life after death

The probability of going to he** is ½, where ** depends on how lean or mean you’ve been to your fellow mortals.

The probability of being reincarnated is 0.

The probability of being resurrected is 0.0000…1.

The probability of being cremated in crowded Singapore is 1/100,000.

Death after life

The probability of living for eternity [heaven or hell] is 1.

The probability of living in purgatory is zero – it’s a cardinal myth.

The probability of being in a Who’s Who in Hell list is exponentially higher than you think.

The probability of being in a Who’s Who in Heaven list is infinitesimally lower than you think.

Born twice, die once – eternal life

Born once, die twice – eternal death

  

The probability of dying twice [physical and spiritual deaths] is 1 × 1/x.

The probability of dying once [physical] is 1 × ½.

The probability of being  “born again” is about 1/15 – based on the number of residents-believers in Singapore.

It’s better to be born twice and die once than to be born once and die twice.

The chances of eternal life depend on your faith in His earthly death.

A lifespan of three scores and ten (70) lasts about a billion seconds.

Would you like to be blessed with a four-scores-and-ten lifespan? 

Then start honoring your parents. 

The probability of living up to 70 years old is 0.7.

The probability of living up to four scores and ten (90) is 0.1.

The probability of living up to five scores and ten (110) is 0.000,000,000,000,1.

On earth he was grounded to infinity

In heaven he was lifted up to eternity

Life is fair

Death is a great equalizer for rich and poor, smart and dumb, strong and weak, arrogant and humble, etc.

For partial proof

Dig six feet deep

Poor Fermat lied about his proof           

So the devil sent it to Prof. Wiles

You and faith = heaven

You and fate = hell

      

Dead Mathematicians: “Stop counting, we ‘re not getting up.”

In his prime, he worked on primes

In eternity, he rested in solitary

Life is the gross common differentiator (GCD) between rich and poor.

Death is the least common denominator (LCM) for all rich and poor.

           

A Christian dies twice but lives once.

An atheist dies once and lives once.

Live truthfully so that your pastor doesn’t have to lie about you at your funeral service.

Atheist: Heaven? No. Hell? Maybe, black hole?

Why are there only a low percent of mathematicians and math educators who go to heaven?

If they all went, it'd be hell!

Eternally yours

© Yan Kow Cheong, May 27, 2010