Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Technologies: Past and Future

January 2000

In a recent online discussion one participant expressed apprehensions that using technology in school would widen the social divide between affluent and poor districts and individual children. Those that could afford better computers and more software would have a superior chance compared to those with limited means. This argument was parried by a remark that it applied in equal degree to other educational tools as well, books being one example. And apparently no case had been reported of objection to using (text)books in school.

I wish to add my two cents. Book is technology no less than computers and the Internet. Although older, it by no means became obsolete nowadays. The current rush to introduce technology into the classroom might be an indication of a general flaw in making good use of technology in the pre-computer days. In the context of education, technology is just a vehicle of presenting and spreading knowledge. Curiosity, a drive to seek and acquire knowledge, is the fuel that lets technology attain its potential.

New technology has a promise to facilitate purposeful search for information. There's no substitute, however, of what is known as reading as applied to books and surfing as applied to the Internet. The term browsing serves in both circumstances.

I am bombarded by the appeals to help with the following puzzle

Sometimes being asked the same question over again is quite annoying. Why on earth do not those people simply look up the answer on the Web? Do they try? Well, I did once just to see how easy or difficult it may be. I searched for various combinations of words triangle, dissection, puzzle, rearrangement, area. It's good I did. True, no search led me to a solution to that puzzle. Instead, I came across a few interesting sites I was unaware existed. ISIS - International Society for the Interdisciplinary Study of Symmetry. Based in Belgrade, the society publishes an online quarterly journal, VISUAL MATHEMATICS. The journal offers artwork and peer reviewed papers. Papers submitted in TEX are converted to HTML by TTH.

Given the puzzle's title, I also tried to search for "Can this be true?" Among a respectable list of irrelevant sites, one contained a jovial story of a fellow who towards the end of a sequence of misfortunes had his buttocks burned and arms broken.

So much for a purposeful search on the Web(*). What about the puzzle? (The credit for the above graphics is probably due to Daniel Takacs whose url was included into one of the inquiries. I located the latter in my mail program's Trash folder searching of all things for "Help" in the subject matter - nothing else worked.)

I usually do not give the answer, but instead refer to some excellent books I wish more people read. Dissections is the topic of the encyclopedic Dissections: Plane & Fancy by Greg Frederickson. (Chapter 23 discusses this and other similar puzzles.) A more popular Mathematics: Magic and Mystery by Martin Gardner has two chapters on geometrical vanishes. I hope that through browsing the book my correspondents will have their curiosity picked beyond a fleeting interest in a trivial puzzle. (However, just in case, the puzzle is known as Curry's paradox, which is discussed elsewhere on this pages.)

This is the nature of browsing - skimming a book or surfing the Web one is up for surprises. Not long ago at an auction I purchased ($3) Martin Gardner's The First Scientific American Book of Mathematical Puzzles and Diversions. What do you know? Chapter 14, Fallacies, talks in particular of a very similar puzzle - the Curry triangle. I'll add the book to the suggested reading list next time I get an inquiry. Meanwhile, while browsing the book, my eye was caught by an integer matrix in Chapter 2, Magic with a Matrix. The puzzle is a generalization of the one I called Calendar Magic.

Select 5 squares, 1 in every column and every row. Sum up the numbers in the selected squares. It's possible to unselect some squares and to select other ones. The matrix below has the wonderful property that selected numbers always add up to the same sum.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at, download and install Java VM and enjoy the applet.

What if applet does not run?

The explanation is very simple. Check the Hint box. A row of numbers will appear at the top and a column of numbers will appear to the left of the matrix. Call them T-controls and L-controls. Each entry of the matrix is the sum of the T-control above it and the L-control to its left. With 5 entries selected 1 per row and 1 per column, the sum of the entries equals the sum of all T- and L-controls. Thus the matrix is far from being unique. More can be produced by modifying control values. (To increase a control value click a little to the right of its central line. To decrease its value click a little to the left. Dragging the cursor will make the number change faster. Control values are in the range from -100 through 100.)

More advanced readers may choose to play with a matrix whose entries are products rather than sums of the controls, in which case five selected numbers must be multiplied. Now it's the product that does not depend on specific selections.

A teacher may think of the matrix as a playful practice activity, a student as a number for a magic math show. A technologically-oriented teacher may want to have a private copy of this and other books because, believe me, it will take a while for the wealth of already published material to migrate to the Web. And even when it does, it won't be that easy to locate it. In any event, it's by far a safer bet to know what is it you are after.

There is another and a more important aspect of the transition from older to newer technology. By itself, say, streaming video or online tutoring will reduce the cost of education. At their best, they will not lower its quality. The Web will help spread the best practice. But the way to develop the best practice is to reach into and master the not so hidden treasures of older technology.

David Jonah from Wayne State University discovered another site that sports the same graphics. The site is by Torsten Sillke, Frankfurt/Main, Germany. The site popped up as the result of a search at for the word disection. To the credit of the google search engine it must be said that it did not get subverted by the misspelling. The search resulted in 811 of annotated links. It began with Interactive Frog Dissection from Unversity of Virginia. The first mathematical reference appeared on page 4 (Dissection Page from the University of Exeter, UK). In between, it was curious to find Fetal Pig Dissection Page, The Exploratorium's Cow's Eye Dissection, Virtual Hospital:The Human Brain. The reference to Geometric Paradox did not appear up to the page #20. I remove my hat to David's perseverance.

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