Information Work: Numbers and Solving Problems

Allen Klinger, © 10/28/2008
Division of space and other issues of great importance in previous eras puzzle most people today. Indeed puzzles pose difficulties. It is easy to become blocked - unable to reason. The idea of this material is to organize a series of puzzling items to support learning activity. (To become qualified for information work, one must get used to overcoming blocks.) One way to describe learning flexibility is thinking outside the box . Puzzles were the domain of an acknowledged master at creating them Sam Loyd - "Loyd produced over 10 000 puzzles in his lifetime many involving sophisticated mathematical ideas." To see his work Begin Here and go to links it describes. A sample is the Horse and Rider puzzle. (Its Solution is available.)


Ascher, M. Ethnomathematics			Kasner/Newman Mathematics & Imagination			Tetrahedron - Four-Sided Block			Big - Large Number Sizes

The organizing principle behind blocks has two parts. First, there's something interesting to learn from a block. As a spiritual song puts it, "Lord, don't take away my stumbling blocks. Just give me the power to climb." Second, an act of will is the essential step in climbing. The act can start by just asking a question such as what does that symbol mean? Two examples are the:
number 8 in the expression 10⁸, an Exponent; and,
exclamation point in 7!, the Factorial.

Using signs and symbols to express ideas is learnable ... if you want to do it.

If you think you can do a thing or think you can't do a thing, you're right. - Henry Ford; Human beings can alter their lives by altering their attitudes of mind. - William James; You may have a fresh start any moment you choose, for this thing we call 'failure' is not the falling down, but the staying down. - Mary Pickford

The thought that led to products - quarrying and fitting stones into walls; things about curves, lines and closeness - endure today. More to the point, we have ability. A try followed by an assist is one way to learn. Many solutions are a click away in other files, such as posted images and book sources.

[A question related to computing (first asked me by Merwyn Sommer).]

Find what should follow 24 in:

10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, __?

This question is usually seen as fairly difficult. But the reason it is so has to do with our inability to think outside the box, i.e., our ordinary mode. Even with a hint (italicized words above), people tend to limit their range of alternatives - in other words, create blocks.

What makes it difficult for most people to answer this? Find a way to make twenty-four given three five numerals, one one number, and the four arithmetic operations, plus, minus, times and divide (visually).
Symbolically and Word Hint; one Solution (with discussion) ; Question .

"Box" Thinking
The following visual puzzle defines for many the concept "think outside the box":
Connect the nine dots by four lines without moving the pencil from the paper

Books list puzzles and explain the many different challenges people have overcome.
Here are four questions to push thought:

1. What is half of eleven? Is an integer answer ever correct??
More Than Five
Binary Variation

2. What is one-half plus one-third? Is there only one true answer??
Baseball Arithmetic
Resistors Etc.

3. Can one arrange nine apples in four baskets so there are an odd number of apples in each basket? What computer principle is involved in solving this problem??
Distribute Apples

4. Can one divide a circular pizza pie into eight exactly equal slices with just three straight line cuts?

Divide Pizza

Table 1. Questions and Web Pointers
The most puzzling thing of all is the relative inaccessiblity of the vast amount of thought human beings have put into quantity, relationships, and logic. Moreover the established names for basic ideas are tricky for the average person. This is explored in some detail in Concepts' Names.
For related drafts please see:

Training/Thinking Visual_Material Association About a Limit Math History About Size

Some of the above concerned numbering. Different number systems are used in computing than in commerce. For material about number systems and counting systems please see Bases. A visually persuasive argument about the Pythagorean theorem can be found from Squares.

Reasoning About a Number Problem

Background and Acknowledgement. This page is in progress. It was initiated 9 Oct 2001. The related drafts describe attempts to address mathematical issues needed for a positive approach to working with computers. Interaction with my students led to material in these drafts. Gavin Wu drew four cartoons. Jean Ji scanned the photograph of an Incan wall from the book by Ascher; for reference detail please click Math History

28 October 2008 Version http://www.cs.ucla.edu/~klinger/blocks.html