A Computer Algorithm for Randomization

Back in the early days of computers, one of the more popular methods of generating a sequence of random numbers was to employ the following scheme: 1. Choose a starting number between 0 and 1. 2. Multiply the starting number by 4 ("stretch" it). Subtract 4 times the square of the starting number from the quantity obtained in step 2 ("fold" the interval back on itself in order to keep the final result in the same range). 3.Given a starting number between 0 and 1, we can use the proce-dure—o...

 How a Nerd Walks Up the Stairs

Your steps should be in a cycle: short, long, long. Long in this case means a double step. Thus, you will cover five stairs in one short-long-long cycle. In addition, you should always start the first cycle on the same foot. Suppose you start on the left foot, then after two cycles you are back on the left foot, having covered ten stairs. While you are walking the stairs in this way, it is clear where you are in the cycle. By the end of the staircase, you will know the number of stairs modulo...

 Constituative Rules of Chutes and Ladders

Players all begin with a value of zero. Players alternate turns adding a random number of 1–6 to their current value. The first player to reach a value of exactly 100 wins (if adding the random number to a player's total would make the total exceed 100, do not add the random number this turn). When a player's total exactly reaches certain numbers, the total changes. For example, if a player reaches exactly 9, her total becomes 31. If a player reaches exactly 49, her total becomes 11.(This rul...

 The Wonder of a Child Learning Their Native Language

Imagine you are faced with the following challenge: You must discover the underlying structure of an immense system that contains tens of thousands of pieces, all generated by combining a small set of elements in various ways. These pieces, in turn, can be combined in an infinite number of ways, although only a subset of these combinations is actually correct. However, the subset that is correct is itself infinite. Somehow you must rapidly figure out the structure of this system so that you c...

 Number Scrabble: Numerical Tic-Tac-Toe

In psychological research on problem-solving, sometimes the game of Tic-Tac-Toe is employed, which, though very simple to learn and play, still offers sufficient problems to the investigator in that it is not at all clear what heuristics are used by the subjects, except avoiding the winning move of the opponent. The same is apparently true for the isomorphic game of Number Scrabble, which is based on the fact that there exists a 3 X 3 magic square, of which rows, columns, and main diagonals a...

 Math Exercise: Multiple Approaches to Problem-Solving

For example, if the problem was to fi nd the answer to 8 × 6, students may suggest three options: memorizing the multiplication table for 6, knowing that 8 × 5 = 40 and adding another 8 to equal 48, or adding a column of six 8s. Allowing students to personally choose among approaches all confi rmed as correct and to support their choice will increase their comfort levels. Th is process also builds math logic, intuition, and reasoning skills that extend into other academic subjects and real-li...