Puzzle for March 12, 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.
Scratchpad
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Hint #1
Add D to both sides of eq.3: E + F + D = C - D + D which becomes eq.3a) E + F + D = C In eq.4, replace C with E + F + D (from eq.3a), and replace A with B + D (from eq.5): E + F + D = B + D + E Subtract both D and E from each side of the above equation: E + F + D - D - E = B + D + E - D - E which makes F = B
Hint #2
In eq.2, replace C with E + F + D (from eq.3a): B + D + F = E + F + D + E Subtract both D and F from each side of the above equation: B + D + F - D - F = E + F + D + E - D - F which simplifies to B = 2×E which also makes F = B = 2×E
Hint #3
In eq.6, substitute 2×E for both B and F: D = 2×E + E + 2×E which makes D = 5×E
Hint #4
Substitute 2×E for B, and 5×E for D in eq.5: A = 2×E + 5×E = 7×E
Hint #5
Substitute 7×E for A in eq.4: C = 7×E + E = 8×E
Solution
Substitute 7×E for A, 2×E for B and F, 8×E for C, and 5×E for D in eq.1: 7×E + 2×E + 8×E + 5×E + E + 2×E = 25 which becomes 25×E = 25 Divide each side by 25: 25×E ÷ 25 = 25 ÷ 25 which means E = 1 making A = 7×E = 7 × 1 = 7 B = F = 2×E = 2 × 1 = 2 C = 8×E = 8 × 1 = 8 D = 5×E = 5 × 1 = 5 and ABCDEF = 728512