**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**

**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 + 2^{1}+ 2^{2}+ 2^{3}+⋯+ 2^{25}.9

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

**. 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, 16

*n*+ 10

*n*– 1 is divisible by 25.

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

(a) SANTA - CLAUS = XMAS

(b) A + MERRY + XMAS = TURKEY

**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 ZerosFor 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.

Now available as an iPad app http://tinyurl.com/7pn8pau |

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