Some Problems to Try
The point of all Professor Tobbs’ work, as you may have guessed, is two-fold at least.
1. Mathematics involves the active construction and application of principles and relationships, not the memorization of inert facts.
As Alfred North Whitehead said almost a century ago:
“In training a child to activity of thought, above all things we must beware of what I call ‘inert ides’ — that is to say, ideas that are merely received into the mind without being utilized, or tested, or thrown into fresh combinations.” (The original essay was published in 1917. See Alfred North Whitehead, The Aims of Education and Other Essays, New York: The Free Press, 1957.)
2. Learning almost always involves provoked adaptation. One constructs new ideas, relationships, hypotheses in the face of moderate mismatch between one’s present state and the demands of a situation.
In the case of (1) and (2) the heart of the issue is a good problem.
Here are some problems I have found engaging and fruitful. Enjoy solving them? Perhaps the list will change from time to time.
1. John loves to divide by 3. He has a basket full of numbers and he takes them out one by one and divides by 3. Sometimes he takes out two at a time and tests each for divisibility by 3 and then he adds them and divides the sum by 3. Sometimes he takes three out and tests each for divisibility by 3, tests the sum of each pair for divisibility by 3, and adds all three and divides the sum by 3. And so forth. Obviously, sometimes the number is divisible by 3 and sometimes it is not. How many numbers does he need to take out randomly from his basket in order that if he looks at them by ones or by sums of two or by sums of three, and so forth, the result is a number divisible by 3?
Generalize.
Ian Harris and Lyndon Baker, Association of Teachers of Mathematics (UK)
2. Consider the sequence 0,1,2,3, ... . Cross out all the numbers which have the digit 3, 6, or 9. Does the 58th remaining number have the digit 2?
Ian Harris and Lyndon Baker, Association of Teachers of Mathematics (UK)
3. BOUGHT COUGH ROUGH THROUGH THOUGHT THOUGH
Any others?
Johnston Anderson, Association of Teachers of Mathematics (UK)
4. You have a wooden cube and two colors of paint, red and white. Each face must be a solid color - no halvies, etc. How many different cubes can be made?
Johnston Anderson, Association of Teachers of Mathematics (UK)
