Puzzle for September 17, 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 D to both sides of eq.2: F – D + D = A + D + D which becomes eq.2a) F = A + 2×D Add A and D to both sides of eq.5: A + F + A + D = C – A – D + A + D which becomes eq.5a) 2×A + F + D = C
Hint #2
In eq.5a, replace F with A + 2×D (from eq.2a): 2×A + A + 2×D + D = C which becomes eq.5b) 3×A + 3×D = C
Hint #3
Add C to both sides of eq.6: B + E – C + C = A + C + D + C which becomes B + E = A + 2×C + D In the above equation, substitute (3×A + 3×D) for C (from eq.5b): B + E = A + 2×(3×A + 3×D) + D which becomes B + E = A + 6×A + 6×D + D which becomes eq.6a) B + E = 7×A + 7×D
Hint #4
In eq.3, substitute 3×A + 3×D for C (from eq.5b), and A + 2×D for F (from eq.2a): 3×A + 3×D + A + 2×D = A + D + E which becomes 4×A + 5×D = A + D + E Subtract A and D from both sides of the above equation: 4×A + 5×D – A – D = A + D + E – A – D which becomes eq.3a) 3×A + 4×D = E
Hint #5
Substitute 3×A + 4×D for E (from eq.3a) in eq.6a: B + 3×A + 4×D = 7×A + 7×D Subtract 3×A and 4×D from each side of the equation above: B + 3×A + 4×D – 3×A – 4×D = 7×A + 7×D – 3×A – 4×D which becomes eq.6b) B = 4×A + 3×D
Hint #6
Substitute 3×A + 4×D for E (from eq.3a), (A + 2×D) for F (from eq.2a), 4×A + 3×D for B (from eq.6b), and 3×A + 3×D for C (from eq.5b) in eq.4: 3×A + 4×D + (A + 2×D) = 4×A + 3×D + 3×A + 3×D – (A + 2×D) which becomes 4×A + 6×D = 7×A + 6×D – A – 2×D which becomes 4×A + 6×D = 6×A + 4×D Subtract 4×A and 4×D from both sides of the above equation: 4×A + 6×D – 4×A – 4×D = 6×A + 4×D – 4×A – 4×D which becomes 2×D = 2×A Divide both sides by 2: 2×D ÷ 2 = 2×A ÷ 2 which becomes D = A
Hint #7
Substitute A for D in eq.6b: B = 4×A + 3×A which makes B = 7×A
Hint #8
Substitute A for D in eq.5b: 3×A + 3×A = C which makes 6×A = C
Hint #9
Substitute A for D in eq.3a: 3×A + 4×A = E which makes 7×A = E
Hint #10
Substitute A for D in eq.2a: F = A + 2×A which makes F = 3×A
Solution
Substitute 7×A for B and E, 6×A for C, A for D, and 3×A for F in eq.1: A + 7×A + 6×A + A + 7×A + 3×A = 25 which simplifies to 25×A = 25 Divide both sides of the above equation by 25: 25×A ÷ 25 = 25 ÷ 25 which means A = 1 making B = E = 7×A = 7 × 1 = 7 C = 6×A = 6 × 1 = 6 D = A = 1 F = 3×A = 3 × 1 = 3 and ABCDEF = 176173