Puzzle for November 20, 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 B and F to both sides of eq.6: C + E – B – F + B + F = A + B + D + B + F which becomes C + E = A + 2×B + D + F which may be written as eq.6a) C + E = A + F + 2×B + D Add D and A to both sides of eq.4: F – D + D + A = C – A + D + A which becomes F + A = C + D which may be written as eq.4a) A + F = C + D
Hint #2
In eq.6a, replace A + F with C + D (from eq.4a): C + E = C + D + 2×B + D which becomes C + E = C + 2×B + 2×D Subtract C from both sides of the equation above: C + E – C = C + 2×B + 2×D – C which becomes eq.6b) E = 2×B + 2×D
Hint #3
In eq.2, replace E with 2×B + 2×D (from eq.6b): D + 2×B + 2×D = A which becomes eq.2a) 2×B + 3×D = A
Hint #4
In eq.3, substitute 2×B + 3×D for A (from eq.2a): B + F = 2×B + 3×D + D which becomes B + F = 2×B + 4×D Subtract B from each side of the equation above: B + F – B = 2×B + 4×D – B which becomes eq.3a) F = B + 4×D
Hint #5
Substitute 2×B + 3×D for A (from eq.2a), and B + 4×D for F (from eq.3a) in eq.4a: 2×B + 3×D + B + 4×D = C + D which becomes 3×B + 7×D = C + D Subtract D from both sides of the equation above: 3×B + 7×D – D = C + D – D which becomes eq.4b) 3×B + 6×D = C
Hint #6
Substitute 2×B + 3×D for A (from eq.2a), 3×B + 6×D for C (from eq.4b), (2×B + 2×D) for E (from eq.6b), and B + 4×D for F (from eq.3a) in eq.5: 2×B + 3×D + 3×B + 6×D – (2×B + 2×D) = B + (2×B + 2×D) + B + 4×D which becomes 5×B + 9×D – 2×B – 2×D = 4×B + 6×D which becomes 3×B + 7×D = 4×B + 6×D Subtract 3×B and 6×D from both sides of the above equation: 3×B + 7×D – 3×B – 6×D = 4×B + 6×D – 3×B – 6×D which simplifies to D = B
Hint #7
Substitute B for D in eq.2a: 2×B + 3×B = A which makes 5×B = A
Hint #8
Substitute B for D in eq.4b: 3×B + 6×B = C which makes 9×B = C
Hint #9
Substitute B for D in eq.6b: E = 2×B + 2×B which makes E = 4×B
Hint #10
Substitute B for D in eq.3a: F = B + 4×B which makes F = 5×B
Solution
Substitute 5×B for A and F, 9×B for C, B for D, and 4×B for E in eq.1: 5×B + B + 9×B + B + 4×B + 5×B = 25 which simplifies to 25×B = 25 Divide both sides of the above equation by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = F = 5×B = 5 × 1 = 5 C = 9×B = 9 × 1 = 9 D = B = 1 E = 4×B = 4 × 1 = 4 and ABCDEF = 519145