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Algorithms for finding prime numbers date back at least as far as ancient Greece, where mathematicians used a straightforward approach known as the Sieve of Erastothenes. The Sieve of Erastothenes works as follows: To find all the primes less than n, begin by writing down all the numbers from 1 to n in sequence. Then…
The 37% Rule derives from optimal stopping’s most famous puzzle, which has come to be known as the “secretary problem.” Its setup is much like the apartment hunter’s dilemma that we considered earlier. Imagine you’re interviewing a set of applicants for a position as a secretary, and your goal is to maximize the…
When dealing with a sequence of length N, the elements of which we wish to distinguish by subscript, the next vexing question is what subscript value to assign to its starting element. Adhering to convention a) yields, when starting with subscript 1, the subscript range 1 ≤ i < N 1; starting with 0, however, gives the…
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…
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…
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…
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…
Buzz. An example of a low-stress, win-win game is Prime Number Buzz. Students stand in a circle or at their desks and go around the room in order, saying either the next sequential number if it is a composite or “buzz” if it is a prime. If they are incorrect, they sit down, but they keep listening and when they catch…
Addition: add, plus, sum, total, altogether, increased by, grew, gained, total of, combined, more than (as in, “3 more than 7 is 10”), put together, in all Subtraction: minus, take away, diff erence, less than, from, remove, subtract, gives away, sells, loses, fewer than, decreased by, diff erence between…
Prepare a list of the calculations from the flash cards on a sheet of paper. These can be on a template, with multiplication facts at the appropriate level pulled and copied for the student. On these forms, include three columns next to each multiplication question, labeled “correct repeat,” “correct wait,” and…
Stanislas Dehaene brings up the Ramanujan-G.H.Hardy anecdote concerning the number 1729. The idea of running through the cubes of all integers from 1 to 12 in order to arrive at Ramanujan's spontaneous recognition of 1729 as the smallest positive integer that can be written in two distinct ways as the sum of two…
The revolution in our understanding of the logic of names began with a basic question: Where do the meanings of words live? There are two likely habitats. One is the world, where we find the things that a word refers to. The other is in the head, where we find people’s understanding of how a word may be used. For…
Children learn very early (often as young as 2 years old) to recite the number-word list in order (Fuson, 1988). But at the beginning, the words are merely placeholders—children recite the list without knowing what the individual number words mean. Over time, children fill in the words with meaning, one at a time and…
The problem is that the laws of addition and multiplication you are using hold for natural numbers, but infinity is not a natural number, so these laws do not apply. If they did, you could use a similar argument that multiplying anything by infinity, no matter how small, gives infinity, thus ∞×0=∞. More sophisticated…
Number is a rich, many-sided domain whose simplest forms are compre- hended by very young children and whose far reaches are still being explored by mathematicians. Proficiency with numbers and numerical operations is an important foundation for further education in mathematics and in fields that use mathematics…
Systems science is best seen as a science that consists of nested sub-sciences. It is presented most compactly using the notation of set theory. Let A represent a science of description. Let B represent a science of design. Let C represent a science of complexity. Let D represent a science of action (praxiology). Let…
Occasionally he made a comment similar to Archimedes' "Eureka!" about a discovery, as he did when he wrong T T T=N. This is part of number theory, the field that Gauss called the Queen of Mathematics. A triangular number T is a number that you can get by adding up a string of consecutive whole numbers beginning with…
Theorem: There are infinitely many prime numbers. Proof: Suppose, contrary to the theorem, that there is only a finite number of primes. Thus, there will be a largest which we can call p. Now define a number n as 1 plus the product of all the primes: n = (2 X 3 X 5 X 7 X 11 X...X p) 1 Is n itself prime or composite?…
The rudest numerical scales, such as that by which the mineralogists distinguish different degrees of hardness, are found useful. The mere counting of pistils and stamens sufficed to bring botany out of total chaos into some kind of form. It is not, however, so much from counting as from measuring, not so much from…
230(231-1) ... is the greatest perfect number known at present, and probably the greatest that ever will be discovered; for; as they are merely curious without being useful, it is not likely that any person will attempt to find a number beyond it.