Puzzle for November 20, 2020 ( )
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
eq.6 may be written as: C = (B + D + E) ÷ 3 Multiply both sides of the equation above by 3: 3 × C = 3 × (B + D + E) ÷ 3 which becomes eq.6a) 3×C = B + D + E Add C, D, and 2×E to both sides of eq.2: B – C – E + C + D + 2×E = C – D + E + C + D + 2×E which simplifies to eq.2a) B + D + E = 2×C + 3×E
Hint #2
In eq.2a, replace B + D + E with 3×C (from eq.6a): 3×C = 2×C + 3×E Subtract 2×C from both sides of the above equation: 3×C – 2×C = 2×C + 3×E – 2×C which makes C = 3×E
Hint #3
In eq.6a, replace C with (3×E): 3×(3×E) = B + D + E which becomes 9×E = B + D + E Subtract E from each side of the above equation: 9×E – E = B + D + E – E which makes eq.6b) 8×E = B + D
Hint #4
Add B to both sides of eq.3: C + D – E + B = A + E + B which may be written as C + B + D – E = A + E + B In the above equation, substitute 3×E for C, and 8×E for B + D (from eq.6b): 3×E + 8×E – E = A + E + B which becomes 10×E = A + E + B Subtract E from both sides: 10×E – E = A + E + B – E which becomes eq.3a) 9×E = A + B
Hint #5
Substitute 9×E for A + B in eq.5: 9×E = D + E Subtract E from each side of the equation above: 9×E – E = D + E – E which makes 8×E = D
Hint #6
Substitute D for 8×E in eq.6b: D = B + D Subtract D from both sides: D – D = B + D – D which makes 0 = B
Hint #7
Substitute 0 for B in eq.3a: 9×E = A + 0 which means 9×E = A
Hint #8
Substitute 8×E for D, 3×E for C, and 9×E for A in eq.4: 8×E + F – 3×E = 9×E + 3×E + E which becomes 5×E + F = 13×E Subtract 5×E from each side of the above equation: 5×E + F – 5×E = 13×E – 5×E which makes F = 8×E
Solution
Substitute 9×E for A, 0 for B, 3×E for C, and 8×E for D and F in eq.1: 9×E + 0 + 3×E + 8×E + E + 8×E = 29 which simplifies to 29×E = 29 Divide both sides of the equation above by 29: 29×E ÷ 29 = 29 ÷ 29 which means E = 1 making A = 9×E = 9 × 1 = 9 C = 3×E = 3 × 1 = 3 D = F = 8×E = 8 × 1 = 8 and ABCDEF = 903818