Puzzle for December 25, 2018 ( )
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
In eq.4, replace A with C - E (from eq.3): C - E + D + E = F which becomes C + D = F In eq.2, replace C + D with F: B = F
Hint #2
Substitute A + D + E for F (from eq.4) in eq.5: C - D + E = D + A + D + E Substitute C - E for A (from eq.3) in the above equation: C - D + E = D + C - E + D + E which becomes C - D + E = C + 2×D Add (D - C) to each side: C - D + E + (D - C) = C + 2×D + (D - C) which simplifies to E = 3×D
Hint #3
In eq.6, replace E with 3×D: A - D = D + 3×D Add D to both sides of the above equation: A - D + D = D + 3×D + D which means A = 5×D
Hint #4
Substitute 5×D for A, and 3×D for E in eq.3: 5×D = C - 3×D Add 3×D to both sides: 5×D + 3×D = C - 3×D + 3×D which means 8×D = C
Hint #5
Substitute 8×D for C in eq.2: B = 8×D + D which makes B = 9×D
Solution
Substitute 5×D for A, 9×D for B and F, 8×D for C, and 3×D for E in eq.1: 5×D + 9×D + 8×D + D + 3×D + 9×D = 35 which simplifies to 35×D = 35 Divide both sides of the above equation by 35: 35×D ÷ 35 = 35 ÷ 35 which means D = 1 making A = 5×D = 5 × 1 = 5 B = F = 9×D = 9 × 1 = 9 C = 8×D = 8 × 1 = 8 E = 3×D = 3 × 1 = 3 and ABCDEF = 598139