By James Tanton
Mathematics Galore! Showcases the very best actions and pupil results of the St. Mark s Institute of arithmetic and invitations you to have interaction the maths your self! experience the pride of deep highbrow play and wonder on the heights to which younger students can upward push. See a few nice arithmetic defined and proved through typical and obtainable means.
Based on 26 essays ( newsletters ) and 8 extra items, Mathematics Galore! deals a wide pattern of mathematical tidbits and treasures, every one instantly attractive, and every a gateway to layers of bizarre intensity and conundrum. choose and skim essays in no specific order and luxuriate in the mathematical tales that spread. Be encouraged in your classes, your math golf equipment and your math circles, or just get pleasure from for your self the bounty of analysis questions and fascinating puzzlers that lie inside of.
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Extra resources for Mathematics Galore! (Classroom Resource Materials)
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W. Benford’s Law from MathWorld-A Wolfram Web Resource. html [WEISSTEIN2] Weisstein, E. W. Equidistributed sequences from MathWorld-A Wolfram Web Resource. html 3 Braids PUZZLER: A Weird Language The language of ABABA uses only two letters, A and B, and any combination of them is a word. ) Also, strangely, a blank space, , is considered a word. The language has the property that any word that ends in ABA or in BAB has the same meaning as the word with them deleted. ) Also, any two consecutive As or Bs can be deleted from a word without changing its meaning.
D) Is there an N such that all integer multiples of N contain the digit 3? We can check that for N = 19507893 each of N , 2N , 3N , . . , 25N contain a three, which answers parts a) and b). Challenge. What is the smallest number that answers a) and the smallest number that answers b)? It is fairly easy to settle part d) in the negative: Lemma 1. There is no positive integer N so that all its multiples, N , 2N , 3N , . . , contain the digit three. Proof. Given a positive integer N we’ll construct a multiple of N that does not contain the digit three.
This means that their difference is a multiple of this number 3n − 3m = 1000k for some k. Thus 3m 3n−m − 1 = 1000k. Since 1000 has no factors in common with 3m , this means that 3n−m − 1 is a multiple of 1000 3n−m − 1 = 1000a for some a Thus 3n−m = 1000a + 1 is a power of three that ends in 001. We can repeat this argument for any of the strings 1, 01, 0001, 00001, and so on. Question. No power of three ends with 002. ) Is there a power of three that ends with 007? The Exponential Nature of the Fibonacci Numbers The newsletter claimed that the sequence of Fibonacci numbers, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .