Puzzle for February 13, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
* BC is a 2-digit number (not B×C).
Scratchpad
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Hint #1
Add B + D + F to both sides of eq.5: C - D + B + D + F = A - B - F + B + D + F which becomes C + B + F = A + D In the above equation, replace A + D with B + E + F (from eq.2): C + B + F = B + E + F Subtract both B and F from each side: C + B + F - B - F = B + E + F - B - F which means C = E
Hint #2
In eq.3, replace E with C: F - C = B - C Add C to both sides of the above equation: F - C + C = B - C + C which makes F = B
Hint #3
eq.6 can be written as: eq.6a) 10×B + C - A = D + E - F Add (D - A) to both sides of eq.5: C - D + (D - A) = A - B - F + (D - A) which becomes C - A = -B - F + D which may be written as eq.5a) C - A = D - B - F
Hint #4
In eq.6a, substitute D - B - F for C - A (from eq.5a): 10×B + D - B - F = D + E - F which becomes 9×B + D - F = D + E - F Add (F - D) to both sides of the above equation: 9×B + D - F + (F - D) = D + E - F + (F - D) which makes 9×B = E which also means C = E = 9×B
Hint #5
Substitute B for F, and 9×B for C in eq.4: B + D + B = 9×B which is the same as 2×B + D = 9×B Subtract 2×B from each side: 2×B + D - 2×B = 9×B - 2×B which means D = 7×B
Hint #6
Substitute 7×B for D, 9×B for E, and B for F in eq.2: A + 7×B = B + 9×B + B which becomes A + 7×B = 11×B Subtract 7×B from both sides: A + 7×B - 7×B = 11×B - 7×B which means A = 4×B
Solution
Substitute 4×B for A, 9×B for C and E, 7×B for D, and B for F in eq.1: 4×B + B + 9×B + 7×B + 9×B + B = 31 which simplifies to 31×B = 31 Divide both sides by 31: 31×B ÷ 31 = 31 ÷ 31 which makes B = 1 making A = 4×B = 4 × 1 = 4 C = E = 9×B = 9 × 1 = 9 D = 7×B = 7 × 1 = 7 F = B = 1 and ABCDEF = 419791