Singapore Math

Sunday, March 31, 2013

The Mathematics of the Cross

Amazing love! How can it be

that thou, my God, shouldst die for me?

The cross isn't just a sacred icon or symbol for religious math educators, but it's also a geometrical shape that has borne much fruit in producing many logic and mathematics puzzles. Few recreational mathematicians, professional and amateur, wouldn't want to be challenged by these fertile dissection puzzles.

Flag of Switzerland

Let's resurrect some of these ancient cross puzzles, which have given so much intellectual satisfaction to those who have decided to take up their cross along their mathematical itinerary. Below are ten of these Christmaths puzzles, which would appeal to above-average elementary and middle-school math students and their teachers.

1. Squares in a Cross

A solid Greek cross can be formed by putting together five cubes, or from a number of squares. How many squares are there?

2. Tessellating and Dissecting Crosses

(a) Show how Greek crosses can form a tessellation.

(b) How can an infinite number of dissections from a cross tessellation produce a square?

3. A matchstick Puzzle

The cross on the left is made up of 19 matches. Move 7 of them to make a pattern consisting of four squares.

4. The Area of a Cross

A cross is made up of five congruent squares. If XY = 10 cm, what is the area of the cross?

5. Cross into Rectangle

Using only two straight cuts, divide the cross on the right into three pieces and reassemble them to form a rectangle twice as long as it is wide.

6. Five-piece Square into Cross

Cut a square into five pieces and rearrange them to form a Greek cross, as shown below.

7. Four-piece Square into Cross

Cut a square into four pieces and rearrange them to form a Greek cross, as shown below.

8. Cross into Hollow Square

The Greek cross on the left has a square-shaped hole in the center.
(a) Rearrange the pieces to make a square that has a hollow cross inside.
(b) Rearrange the pieces so that the resulting figure is a square that is rather smaller than the previous "hollow" one.

9. The Cross and the Crescent

Reassemble the seven pieces of the crescent to make the Greek cross.

10. The Rolling Disc

In the figure below, each side of the cross is 10 cm long. A small circular disc of radius 1 cm is placed at one corner. If the disc rolls along the sides of the figure and returns to the starting position, find the distance traveled by the center of the disc.

Selected answers/solutions

1. 22 squares.
4. 100 cm².
5.

10. (104 + 2π) cm

Reference
Yan, K. C. (2011). CHRISTmaths: A creative problem solving math book. Singapore: MathPlus Publishing.

© Yan Kow Cheong, March 31, 2013.

Code: WR3ZTVKVUW9Z

Monday, March 18, 2013

31 Reasons to Smile

1. A Facebook invitation to connect with a long-lost friend thousands of miles away—across the other side of the continent.

2. Finding geometric shapes in the clouds.

3. The smell of a math book fresh from the printer.

4. Seeing someone helping a blind 70-year-old crossing the road.

5. Your child sneaking in her phone even when she's answering to nature's call.

6. Your daughter undecided whether she'd go out with that boy from her class.

7. Double rainbows.

Double Rainbow
© www.vibrantnation.com

8. Hearing that a friend or colleague has successfully gone through her chemotherapy sessions, and she's recovering fast.

9. The public library opens on Boxing Day.

10. Finding the clue to solving a mathematical problem.

11. Going to a public forum and finding that the free food tastes better than those served at five-star hotels.

12. Middle-aged women with heavy make-ups and dyed hair, and they look pretty ugly!

13. Men and women who wear winter clothes in humid, hot Singapore.

14. Your hamster or guinea pig which greets you when you come back home.

15. When you find that you're the only one using the toilet, or the only passenger in a public bus.

16. Ordering a dessert you hadn't had for ages at a food carnival.

17. A free copy of a newly published book from an author-friend himself, which you plan to buy at your next visit to the bookstore.

Junk Mail
© www.toonpool.com

18. Grammatically correct spam e-mails, where the language proficiency is better than those sent by your editors.

19. Being able to access your e-mail at a location you thought would not receive a Wi-Fi signal.

20. Preachers who blushed on expounding on topics such as sex, lying, and secret sins.

21. Hearing someone fart loudly in a train or in a public library.

22. Seeing semi-naked men whose breasts are larger than those of most women.

23. An old folk nervously solving a sudoku.

24. Receiving those politically correct rejection letters, saying that your skills don't match the requirements of the post applied.

25. The seat next to you staying empty on a plane.

26. Shopkeepers who use the abacus to find the total amount you need to pay.

27. Remember composing your first SMS or e-mail.

28. Finding an out-of-print math book at a second-hand bookstore.

29. Spotting a slide rule or logarithm table at a flea market—vintage math.

A typical 20th century slide rule
© cai.cam.ac.uk

30. Your average grade seven math student giving private tuition to a grade two student.

31. Seeing your child solve an elementary math problem that you couldn't solve when you're at her age.

© Yan Kow Cheong, March 18, 2013

Monday, March 4, 2013

The Dogs-and-Ducks Problem

Recently, while reviewing some grade 5 olympiad questions for a local company, which specializes in conducting olympiad math programs in elementary schools, I came across the following nonroutine dogs-and-ducks question.  

The number of ducks is 10 fewer than the number of dogs.
The dogs have three times as many legs as the ducks.
How many ducks and how many dogs are there?  

When I realized that my mental calculated answer differs from the one given by the problem poser, I thought maybe my incorrect solution might serve as a "logic exercise" to tickle others' mathematical bones to point out the flawed reasoning. My faulty argument followed something along these lines:

Since there are 10 fewer ducks, and each duck has 4 legs, there are 10 × 4 = 40 fewer duck legs.  

From the model drawing,  

2 units = 40 

1 unit = 20  

Number of duck legs = 1 unit = 20 

Number of ducks = 20 ÷ 2 = 10  

Number of dog legs = 3 units = 3 × 20 = 60 

Number of dogs = 60 ÷ 4 = 15  

A quick check shows that the answers don't satisfy the given conditions in the question. Although the dogs have three times as many legs as the ducks, however, the number of ducks is only 5 fewer than the number of dogs, which is expected to be 10.  

When the why is harder than the how  

The lesson here is to identify the illogical step, although seasoned problem solvers would probably not commit this kind of error or blunder.   In general, it's one thing not to make an error, and get the correct answer; it's another thing to be able to explain someone else's error.

At a deeper level, it's sometimes conceptually harder to explain someone's flawed reasoning to an incorrect answer than just know how to solve a problem. Think of mathematical fallacies and paradoxes, where in a number of cases the illegal step is anything but obvious.  

A fabricated faux solution

Alternatively, if we argued that a dog has 2 more legs than a duck, we'd then have a model that looks like this:

In this case, students would easily realize that the answers are also incorrect, when they find out that the number of dogs isn't an integer.

Two quick-and-dirty solutions

Why don't you try solving the dogs-and-ducks question on your own first, before comparing your solution with the ones I've worked out? If yours is an alternative (or elegant) solution, the mathematical brethren couldn't wait to reading it!

Method 1

Since a dog has twice as many legs as a duck, and there are 3 times as many dog legs as duck legs, there are 3/2 times as many dogs as ducks.

A model depicting the above relationship may be drawn as follows:

From the model drawing,

1 unit = 10

Number of ducks = 2 units = 2 × 10 = 20

Number of dogs = 3 units = 3 × 10 = 30

Check: Ducks: 20 × 2 = 40 legs

Dogs: 30 × 4 = 120 legs = 3 × 40 legs

Method 2

Given: Number of dogs = Number of ducks + 10

Number of duck legs = 2 × Number of ducks

Number of dog legs = 4 × Number of dogs

Given: Number of dog legs = 3 × Number of duck legs

A quick-and-dirty model representing the above information may be drawn as follows:

From the model drawing,

3 × 2 units = 6 units = 4 units + 10 + 10 + 10 + 10

2 units = 10 + 10 + 10 + 10

1 unit = 10 + 10 = 20

1 unit + 10 = 30

Therefore, there are 20 ducks and 30 dogs.

Method 3 (Sakamoto method)

Those of you who are versed with the three-step Sakamoto method in solving word problems, may proceed as follows:

1. Grasp the relation

Let ④ represent the number of duck legs.

Duck legs Dog legs

④ ⑫

__________________________

Number of ducks Number of dogs

④ ÷ 2 = ② ⑫ ÷ 4 = ③

– 10

__________________________

1 : 1

~~+ 10~~

2. Diagram

3. Number sentences

③ – ② = ① = 10

② = 2 × 10 = 20 (ducks)

③ = 3 × 10 = 30 (dogs)

The number of ducks is 20, and the number of dogs is 30.

Two bonus problems

If you like being challenged by similar dogs-and-ducks or chickens-and-rabbits problems, may I direct you to solving two more questions at Yan's One Minute Math Blog?

Happy Mathematical Problem Solving!

© Yan Kow Cheong, March 3, 2013

Tuesday, February 12, 2013

How to Choose Your Marriageable Date

If you were of marriageable age and weren't superstitious, how would you go about choosing your wedding date, one that would help you to remember your wedding anniversary for a long time? One rational, yet free, approach is to choose a “marriageable date.”

A marriageable date may be defined as one that is based on some simple numerical pattern, which a couple can easily remember long after the wedding day is over.

Socially speaking, marriageable dates ought to be religion-free and race-free—they should be derived from some easily recognizable patterns.

Economically speaking, it doesn’t cost more than a few drops of ink to write the memorable date, free from any numerological calculation or influence of a fortune-teller or New Age charlatan.

Sociologically speaking, tying the knot on a marriageable date helps you to divorce yourself from any irrationality and superstition that certain dates in a given year are more auspicious than others.

Palindromic Dates

A palindrome is a word that spells the same when read forward or backward. For instance, MADAM and EVE. And some palindromic sentences or phrases that sound mathematical are

I prefer pi. Never odd or even

In mathematical parlance, a palindromic number is one that reads the same when its digits are reversed. Examples of palindromes are 101, 313, 10301, 23432, and 3535353.

Now, let's look at some palindromic dates, the British way:

11th November 2011: 11/11/11 11th February 2011: 11/02/2011

21st November 2012: 21/11/12 21st February 2012: 21/02/2012

31st March 2013: 31/3/13 31st August 2013: 31/8/13

@ NewYorker: A palindrome cartoon by Paul Karasik

Prior to 2013, we had "hot dates," such as the following:

20th November 2012: 20/11/2012 or 20112012

20th December 2011: 20/12/2011 or 20122011

20th December 2012: 20/12/2012 or 20122012

What about 20th December 2013? 20/12/2013 or 20122013

Other patternful or pseudo-patternful dates in 2013 are:

1/12/13 2/11/13 12/1/13 9/11/13

13/2/13 13/11/13 11/12/13 11/2/13

The Beasted Date

On 6th June 2006, the date was “060606”—the beasted date that would have made some hardcore believers sick, sick, sick! But in Singapore, it was considered an “auspicious" date which led 310 couples to tie the knot at the Registry of Marriages (ROM). And in Penang, Malaysia, 89 couples also took advantage of that allegedly “lucky” date.

On 3rd March 2003, the date “030303” drew 163 weddings. Based on the DDMMYY pattern, there was also a record of couples tying the knot on 12 December 2012.

The Friday the Thirteenths

Another group of quasi-marriageable dates that superstitious couples usually avoid are those that fall on a Friday the 13th. Two such “unlucky” dates in 2013 that rational folks could leverage on to book their D-Days are:

13th September: 13/9/13

13th December: 13/12/13

The Ghost Month—a Most Marriageable Month

Inauspicious or ill-omened dates superstitious Chinese couples shy away occur during the Seventh Lunar Month, or Ghost Month. This is when rational and budget-conscious couples ought to hold their weddings—when most hotels are charging patrons a promotional rate. Besides, since many irrational or fearful couples avoid tying the knot during the seventh month, the saving for holding a grand wedding at a five-star hotel, at a reasonable price, on any of those "unlucky" days, can be quite significant.

Dates with Consecutive Numerals

One set of marriageable dates involves having the day, month, and year in consecutive order:

8th July: 08/07/06 9th August: 09/08/07

13th December: 13/12/11 11th December: 11/12/13

Odd- and Even-numbered Dates

Another set of hot wedding dates have the day, month, and year, in increasing or decreasing multiples of a whole number.

8th October: 8/10/12 9th November: 9/11/13

7th October: 7/10/13 4th August: 4/8/12

3rd August: 3/8/13 14th December: 14/12/10

How many such marriageable dates are there in the Lunar Year of the Snake?

D-Day on Valentine's Day

A record 1080 couples registered their marriage on Valentine's Day in 1995, as the Chinese considered it a Double Happiness Day—it coincided with the Chinese version of the Valentine's Day, also known as yuan xiao jie, which is on the 15th day of the first month of the Chinese New Year. What's your prediction on February 14, 2013?

Symbolically, tying the knot on Valentine's Day may prove memorable, but unaffordable to many couples as they look for a wallet-friendly hotel to hold their wedding and their subsequent wedding anniversaries.

Date and Time Included

If we marry the time with the date, more interesting patterns emerge. For example, on 4th May 2006 at 8 minutes and 9 seconds, after 7am, the date and time spelled out as

04. 05. 06. 07. 08. 09

Or, on 5th June 2007 at 9 minutes 10 seconds past eight, we had:

05. 06. 07. 08. 09. 10

Can you think of some memorable dates in 2013 and beyond, following a similar pattern?

British (UK) versus American (US) Dates

All the above patternful dates and times are based on the British way of writing dates and times. If we write dates and times the American way, a new set of marriageable dates are generated.

In British English, dates are generally written with no comma between the month and year: 7 May 2013. American English prefers the style May 7 2013 or May 7, 2013.

In numerical notation, there is a difference: 7/5/13 means 7 May 2013 in Britain and 5 July 2013 in America.

So, decide in advance whether you want to wed in a country where they follow the British or American system of writing numerals. A rational option may save you a few hundred or thousand bucks!

Conclusion

The beauty and utility of marriageable dates is that they don't have to tie up with astrology or geomancy to be sanctioned—they're free from any pseudoscientific influences, which would save you, at the least, a few hundred or thousand dollars and months of headache. Marriageable dates are based purely on meaningful patterns; besides, they provide no excuse for any party in future to forget their wedding anniversary. Indeed, nothing is left to chance and randomness, or fear and superstition.

Patternful dates are more interesting and memorable than geomancized or numerological dates. In addition, a by-product of marriageable dates can prove mathematically educational, as compared to feng shui-formulated dates, which are credulously auspicious and costly. One is based on logic and reason; the other is based on irrationality and fear. So, let's marry reason and patterns to choose our wedding date, while divorcing ourselves from any superstition and irrational fear.

Questions

1. How many palindromic “marriageable dates” are there in 2013, if they are written in both British and American English?

2. A set of hot wedding dates have the month, day, and year, in ascending or descending order. For example, 9/11/13 and 12/10/8. How many such odd- and even-numbered dates are there in 2013, if dates are written the American way?

3. On October 11, 2012, at 8 minutes and 9 seconds after 7 o’clock, some Americans expressed the time as follows: 07. 08. 09. 10. 11. 12. How many similar dates can be expressed in this century? How would the answer change if the dates were written the British way?

4. Who would be "luckier," by having more "marriageable dates" in a typical year? Those who write their dates the American or British way? Explain.

References

Deng, Fern (2006). Devilish or not, 666 is hot wedding date. The Straits Times, June 7, 2006, p. H4.

Koh, Melody (2006). 06.06.06 was a lucky date for them. Today, June 7, 2006, p. 6.

A devil of a date. The Straits Times, June 5, 2006, p. 12.

'030303' draws 163 weddings. Streats. March 3, 2003, p. 2.

Singapore Math

Welcome to K C Yan's Singapore Math blog!

Singapore Math

Sunday, March 31, 2013

The Mathematics of the Cross

Monday, March 18, 2013

31 Reasons to Smile

Monday, March 4, 2013

The Dogs-and-Ducks Problem

Tuesday, February 12, 2013

How to Choose Your Marriageable Date

About Me

Blog Archive