Puzzle for December 3, 2021 ( )
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 C to both sides of eq.4: D + E – C + C = B + C + C which becomes eq.4a) D + E = B + 2×C Add E to both sides of of eq.3: B – E + E = C + D + E which becomes eq.3a) B = C + D + E
Hint #2
In eq.3a, replace D + E with B + 2×C (from eq.4a): B = C + B + 2×C which becomes B = B + 3×C Subtract B from both sides of the above equation: B – B = B + 3×C – B which becomes 0 = 3×C which means 0 = C
Hint #3
In eq.2, replace C with 0: 0 + F = A which makes F = A
Hint #4
In eq.3a, replace C with 0: B = 0 + D + E which makes eq.3b) B = D + E
Hint #5
In eq.5, replace C with 0: A + 0 – D = D + E which becomes A – D = D + E Add D to both sides of the above equation: A – D + D = D + E + D which becomes eq.5a) A = 2×D + E
Hint #6
eq.6 may be written as: B + E = (A + C + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × (B + E) = 3 × ((A + C + F) ÷ 3) which becomes eq.6a) 3×B + 3×E = A + C + F
Hint #7
In eq.6a, substitute 0 for C, and A for F: 3×B + 3×E = A + 0 + A which becomes eq.6b) 3×B + 3×E = 2×A
Hint #8
Substitute (D + E) for B (from eq.3b), and (2×D + E) for A (from eq.5a) in eq.6b: 3×(D + E) + 3×E = 2×(2×D + E) which becomes 3×D + 3×E + 3×E = 4×D + 2×E which becomes 3×D + 6×E = 4×D + 2×E Subtract 3×D and 2×E from both sides of the equation above: 3×D + 6×E – 3×D – 2×E = 4×D + 2×E – 3×D – 2×E which makes 4×E = D
Hint #9
Substitute (4×E) for D in eq.5a: A = 2×(4×E) + E which becomes A = 8×E + E which makes A = 9×E and also makes F = A = 9×E
Hint #10
Substitute 4×E for D in eq.3b: B = 4×E + E which makes B = 5×E
Solution
Substitute 9×E for A and F, 5×E for B, 0 for C, and 4×E for D in eq.1: 9×E + 5×E + 0 + 4×E + E + 9×E = 28 which simplifies to 28×E = 28 Divide both sides of the above equation by 28: 28×E ÷ 28 = 28 ÷ 28 which means E = 1 making A = F = 9×E = 9 × 1 = 9 B = 5×E = 5 × 1 = 5 D = 4×E = 4 × 1 = 4 and ABCDEF = 950419