Monthly Archives: June 2014

Puzzle | quadruplet (1, 22, 41, 58)

        Consider the quadruplet   (a, b, c, d) = (1, 22, 41, 58) the sum of any three of them is a perfect square:     1 + 22 + 41 = 64     1 + 22 + 58 … Continue reading

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Powers of 2 puzzle

    Let M be a positive integer. Rearranging the digits of M, we obtain the positive integer N. No leading zeros. Is it possible to have two different positive integer powers of 2 this way?         … Continue reading

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Puzzle | Solve a + b + c + d = n √(abcd) (positive integers)

    I like this number 1281: Concatenation of   3*4 || 3^4, DigitSum(1281) = 12. First two digits:   12 1+2+8+1 = 3√(1*2*8*1)                  

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Puzzle | Factoring the Difference of Two Squares

                                                                   ——————————————                                          

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(1^3 + 3^3 + 5^3 + …+(2n-1)^3)/(1 + 3 + 5 +…+(2n-1))

                     

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Heronian triangle| Area = k*(Perimeter), k is a prime number

  The lenghts of the sides of an isosceles triangle are integers, and its area is the product of the perimeter and a prime. What are the possible values of the prime? Answer:     2,   3,   5   … Continue reading

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Puzzle | 8-digit Num3ers

    How many 8-digit numbers are there in which every digit occurs the same number of times as the value of the digit?   e.g.   3414434,   digit 1 occurs 1 time,   digit 3,   3 times, … Continue reading

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k consecutive integers divisible by the first k prime numbers

                               

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Integer N such that N = DigitSum (N^n), n = 3,4,…,10

        Determine the smallest integer N   with N = DigitSum (N^n),   n > 10 For example,     Here’s a large one: 2015^137,   DigitSum(2015^137) = 2015 Any others?             … Continue reading

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When N – DigitSum(N) and N + DigitProduct(N) are both squares

      Republic of Math @republicofmath says: in base 3 :   2 , 7 , 1457 in base 9 :   2, 8, 24, 73, 80, 161, 265, 1313, 2719, 38446               … Continue reading

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