CTK Insights

The Bottleneck Principle The Bottleneck Principle is a problem-solving strategy according to which it may be useful to look into the circumstances in which the conditions of a problem at hand are either hardly or not at all satisfied. It is different from the Worst-Case Scenario in that the latter looks at the problem as [...]

The Extreme Principle The Extreme Principle is a misnamed problem-solving tactic akin to the Worst-Case Scenario often used in combinatorics and computer science. It does not make any claim (like, say, the Pigeonhole Principle) per se, but only suggests that, for some problems, looking into extreme circumstances or elements within the conditions of the problem [...]

This puzzle comes from a wonderful Russian site, where its solution is presented as a sequence of animations. Is it possible to wrap the cube with a 3×3 piece of paper below it? Handling of the paper is subject to two conditions: The paper may be only cut or folded along the crease lines. The [...]

Nowadays, finding the area of curvilinear shapes falls in the purview of calculus. But the problem of finding areas draw much interest in antiquity and preoccupied mathematicians ever since. One of the acknowledged results by Hippocrates of Chios (470-410 B.C.) is the Squaring of a Lune. The problem of squaring a shape refers to a [...]

The setup A cube whose faces have been partitioned by the midlines into four squares. The activity Color the squares in as many colors as you wish, with a single caveat: no two adjacent squares, i.e., squares that share an edge, may be of the same color. The task Determine the maximum possible number of [...]

What is the setup? The setup for this activity is the graph with 8 nodes joined as shown: What is the task? The task is to place the integers 1 through 8 onto the nodes of the graph so that no two successive integers are "graph neighbors", i.e., joined by an edge. What to observe? [...]

The setup Students are seated in a circle. The number of students may be arbitrary, but in a small group it is much easier to discern the idea behind this activity. With a large number of students, it is preferable to form several groups of about 4-5 kids than to have a single big one. [...]

Much of problem solving in mathematics is about finding a representation in a way that simplifies if not trivializes a given problem. Come to think of it, putting a word problem into algebraic terms - as an equation or a system of equations - is ultimately finding another representation of the problem, a representation more [...]

Breaking chocolate bars is one of my most favorite activities. Assume you have a chocolate bar consisting, as usual, of a number of squares arranged in a rectangular pattern. Your task is to split the bar into small squares (always breaking along the lines between the squares) with a minimum number of breaks. How many [...]

For Day 4 of the break, I shall present 4 games that may at first appear rather unrelated; this is the purpose of the activity to discover the commonality between the four. All four are two-players games, with the players moving in turns. The Fish Soup game For the game you should prepare 9 cards [...]