Monthly Archives: November 2015

Divisibility puzzle : a*b*c | (a + b + c)^n, n = 7,13,21,31

  Definition : If   a   and   b   are integers (with   a   not zero),   we say   a divides   b if there is an integer   c   such that   . we … Continue reading

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Game theory | The Hunter’s Share

    This game is a little more life-like than the Prisoner’s Dilemma, as it involves more than two protagonists. The aim is to gain the most points, or ‘food’, after ten rounds or so. Imagine you are one of … Continue reading

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Num3er 11296321

      is a perfect square containing precisely one 3, two 2’s and three 1’s. Any of the other seven digits   (0, 4, 5, 6, 7, 8, 9)   occur only once.     Find other examples.   … Continue reading

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Oblong numbers as a sum of two squares

  A pronic number is a number which is the product of two consecutive integers, that is, a number of the form   . They are also called oblong numbers.   Prove that there are infinitely many positive integers   … Continue reading

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y = C(n, 7) – floor(n/7)

    Show that… if   then,    is divisible by 7         is the largest integer not greater than   x   and   is the smallest integer not less than   x   is the … Continue reading

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a + b + c + a*b + b*c + c*a = a*b*c + 1

    Find all triples     of positive integers such that ,     and         Solution: (a, b, c)   =   (2, 4, 13),   (2, 5, 8),   (3, 3, 7)       … Continue reading

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Four consecutive terms of an AP

  We know that if you multiply any four consecutive positive integers and add 1 to the product, you’ll get a square number.   The product of four consecutive terms of an arithmetic progression added to the fourth power of … Continue reading

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y = (x!)^1/x

    seems to be approaching a linear asymptote.   Find an approximation for the equation of this line.                  

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Grid 3×3 – Num3ers & Letters

    5     22     18 28     15      2 12      8     25   rows: 5+22+18   =   45 28+15+2   =   45 12+8+25   =   45 columns: 5+28+12   =   45 … Continue reading

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