Puzzle for September 12, 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
Subtract the left and right sides of eq.4 from the left and right sides of eq.2, respectively: D + F – (E + F) = A + E – (A + C) which is equivalent to D + F – E – F = A + E – A – C which becomes D – E = E – C Add E and C to both sides of the above equation: D – E + E + C = E – C + E + C which becomes eq.2a) D + C = 2×E
Hint #2
eq.3 may be written as: B = D + C In the above equation, replace D + C with 2×E (from eq.2a): B = 2×E
Hint #3
Subtract E from both sides of eq.2: D + F – E = A + E – E which becomes eq.2b) D + F – E = A In eq.1, substitute (D + F – E) for A (from eq.2b), and 2×E for B: (D + F – E) – D + F = 2×E – (D + F – E) which becomes 2×F – E = 2×E – D – F + E which becomes 2×F – E = 3×E – D – F In the equation above, add E and D to both sides, and subtract 2×F from both sides: 2×F – E + E + D – 2×F = 3×E – D – F + E + D – 2×F which simplifies to eq.1a) D = 4×E – 3×F
Hint #4
Substitute 4×E – 3×F for D (from eq.1a) in eq.2b: 4×E – 3×F + F – E = A which becomes eq.2c) 3×E – 2×F = A
Hint #5
Subtract D from both sides of eq.2a: D + C – D = 2×E – D which becomes C = 2×E – D In the above equation, substitute (4×E – 3×F) for D (from eq.1a): C = 2×E – (4×E – 3×F) which is equivalent to C = 2×E – 4×E + 3×F which becomes eq.2d) C = 3×F – 2×E
Hint #6
Substitute 2×E for B, and 3×F – 2×E for C (from eq.2d) in eq.5: (2×E + 3×F – 2×E) ÷ F = E which becomes (3×F) ÷ F = E which makes 3 = E making B = 2×E = 2×3 = 6
Hint #7
eq.6 may be written as: F = (B + C + D) ÷ 3 Multiply both sides of the equation above by 3: 3 × F = 3 × (B + C + D) ÷ 3 which becomes eq.6a) 3×F = B + C + D
Hint #8
In eq.6a, substitute 6 for B, 3×F – 2×E for C (from eq.2d), and 4×E – 3×F for D (from eq.1a): 3×F = 6 + 3×F – 2×E + 4×E – 3×F which becomes 3×F = 6 + 2×E Substitute 3 for E in the equation above: 3×F = 6 + 2×3 which becomes 3×F = 6 + 6 which makes 3×F = 12 Divide both sides by 3: 3×F ÷ 3 = 12 ÷ 3 which makes F = 4
Hint #9
Substitute 3 for E, and 4 for F in eq.2c: 3×3 – 2×4 = A which becomes 9 – 8 = A which makes 1 = A
Hint #10
Substitute 4 for F, and 3 for E in eq.2d: C = 3×4 – 2×3 which becomes C = 12 – 6 which makes C = 6
Solution
Substitute 3 for E, and 4 for F in eq.1a: D = 4×3 – 3×4 which becomes D = 12 – 12 which makes D = 0 and ABCDEF = 166034