Problems with Multiplicative Magic Square?

1. In a multiplicative magic square of order 3 (see illustration below), the product of the three entries in each line (row, column, diagonal) are equal to a magic constant Lamda. If all the entries in a magic square of order 3 are positive integers, what are all the possible values for the magic constant lamda that are in the interval [1000, 2008]?

a b c

d e f

g h i

2. The figure below is a multiplicative magic square. If all the entries are positive integers, what is the sum of all the possible values of g?

50| b | c

---+--+---

.d | e | f

---+--+---

.g | h | 2

1 Answer

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  • 1 decade ago
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    1) abc = def = ghi = adg = beh = cfi = aei = ceg = λ

    Take all products containing e,

    def = beh = aei = ceg = λ

    df = bh = ai = cg = λ/e

    But

    acgi = abc * ghi / (beh) * e = λe

    acgi = ai * cg = λ^2/e^2

    λe = λ^2/e^2

    e^3 = λ

    Thus λ must be a cube of a positive integer number. There are only three such numbers in [1000, 2008]:

    10^3 = 1000

    11^3 = 1331

    12^3 = 1728.

    2) Using the previous result,

    bh = cg = ai = 2*50 = 100

    abcghi = abc * ghi = λ^2

    abcghi = ai * bh * cg = 100^3 = 10^6

    => λ = 10^3 = 1000

    Now immediately

    e = 10

    c = 100/g

    d = 20/g

    h = 500/g

    so g must be a divisor of gcd(500,100,20) = 20 for all of these to be integer. Let's continue to find whether there are further constraints,

    f = 100/d = 5*g

    b = 100/h = g/5

    so g must also be a multiple of 5.

    Therefore, the only possible values of g are 5, 10 and 20. Their sum is 35.

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