## Welcome to K C Yan's Singapore Math blog!

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## Saturday, December 17, 2011

### 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) × end in 25 zeros?

A Merry CHRISTmaths to all!

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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## Thursday, December 15, 2011

### 25 Things You Wished Were Untrue about Christmas

25. Toys are made affordable for millions of parents thanks to the appalling working conditions endured by under-aged children in Santa’s sweatshops, in China.

24. Proud CEO: “My time in looking for a five-dollar present for my company Christmas party is worth more than US\$2,525—I’d rather issue a check 100 times the amount for the gift.”

23. An e-card costs much less than 25 cents—maybe a mere 2.5 cents plus or minus a few bits and bytes.

22. There are fewer joyful people on Christmas Day than on any other day of the year. Do you know why?

21. Men dishonor Christ more in the 12 days of Christmas than in all the 12 months of the year—by their boozing and carousing.

20. Even staunch Christians may not be aware that Christmas is a pagan feast, originally meant to worship the Sun god, instead of the Son of God.

19. There may be more non-believers celebrating Xmas than believers every year.

18. Singles, poor people, and entrepreneurs could not wait for Christmas to come and go—not a happy time to be alone or be lonely.

17. Many believers and non-believers celebrate Christmas without a genuine understanding of the meaning of the public holiday—probably the most celebrated and the least understood festival of the year.

16. Retailers rely on Xmas significantly on the festive sales to remain financially afloat.

15. People merry so much on Xmas Eve that they don’t have time to open up their Xmas gifts—Boxing Day often becomes a sleeping day!

14. Even believers forget that Christmas needn’t be merry to be meaningful—it is the Christ of Christmas that needs to be celebrated, not Christmas itself.

13. Believers and non-believers are more and more expensively anxious or stressed to buy something to please others—as a sign of showing off their wealth, or exhibiting misplaced generosity.

12. New Christians do not want to be excluded from Christmas as they await for their employers’ Christmas bonus—that universally revered figure.

11. Employees are willing to work under unreasonable bosses because of the Christmas bonus—many at the same place, for the same boss, for more than 25 years.

## The Search

10. Believers from other faiths often feel morally obliged to celebrate Christmas with their Christian colleagues and neighbors, all in the name of religious harmony.

9. Religious fanatics intolerant of Christianity wished Christmas were celebrated every quarter of a century instead of every year.

8. Some people would buy off a Christmas gift for their children on eBay for several thousand times the original price.

7. People max out their credit cards to finance the gift storm, as compared to their grandparents who saved money for Christmas.

6. Some allegedly US-friendly Arab nations tend to treat Christians worse than those living under dictatorial Muslim rule.

5. A month after Christmas, the holiday is only halfway paid off—credit card debt resulting from a borrowed merry Christmas. And a third of the money borrowed for Christmas spending is still not paid off two months after the holiday.

 A Christmas gift for numbers lovers http://tinyurl.com/7pn8pau
4. This Christmas over a billion people in the world are surviving on less than 70 pence a day.

3. Politically correct retailers have banished Jesus and Christmas from the mall to accommodate to the secular demands of agnostics and non-theists.

2. Christmas is celebrated today more like a sales frenzy than as the most important birth in history—the commercial mentality has robbed the message of giving.

1. The closer people get to Christmas, the pressure to give these (useless) unnecessary gifts goes up and they feel depressed and unworthy if they can’t give.

Merry X-W/g to all!

© Yan Kow Cheong, Dec. 15, 2011