Puzzle for October 30, 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
In eq.2, replace F with A + B + D (from eq.1): A + B + D – A = A + B which becomes B + D = A + B Subtract B from each side of the equation above: B + D – B = A + B – B which makes D = A
Hint #2
In eq.3, replace D with A: C – A = A + B – C + A + F which becomes C – A = 2×A + B – C + F Add A and C to both sides of the above equation: C – A + A + C = 2×A + B – C + F + A + C which becomes eq.3a) 2×C = 3×A + B + F
Hint #3
Add A to both sides of eq.2: F – A + A = A + B + A which becomes eq.2a) F = 2×A + B eq.3a may be re-written as: eq.3b) 2×C = A + 2×A + B + F
Hint #4
In eq.3b, substitute F for 2×A + B (from eq.2a): 2×C = A + F + F which becomes 2×C = A + 2×F Subtract 2×F from each side of the equation above: 2×C – 2×F = A + 2×F – 2×F which becomes 2×C – 2×F = A which may be written as eq.3c) 2×(C – F) = A
Hint #5
Substitute D ÷ B for C – F (from eq.4) into eq.3c: 2×(D ÷ B) = A which is equivalent to (2×D) ÷ B = A Multiply both sides of the above equation by B: ((2×D) ÷ B) × B = A × B which becomes eq.3d) 2×D = A × B
Hint #6
Substitute A for D in eq.3d: 2×A = A × B Since A ≠ 0 (from eq.5), divide both sides of the above equation by A: 2×A ÷ A = (A × B) ÷ A which makes 2 = B
Hint #7
Substitute 2 for B in eq.2a: eq.2b) F = 2×A + 2 Substitute 2 for B, and 2×A + 2 for F (from eq.2b) in eq.3a: 2×C = 3×A + 2 + 2×A + 2 which becomes 2×C = 5×A + 4 Divide both sides of the above equation by 2: 2×C ÷ 2 = (5×A + 4) ÷ 2 which becomes eq.3e) C = 2½×A + 2
Hint #8
eq.6 may be written as: C – F = (A + E) ÷ 2 Substitute 2½×A + 2 for C (from eq.3e), and (2×A + 2) for F (from eq.2b) in the equation above: 2½×A + 2 – (2×A + 2) = (A + E) ÷ 2 which is equivalent to 2½×A + 2 – 2×A – 2 = (A + E) × ½ which becomes ½×A = ½×A + ½×E Subtract ½×A from both sides of the above equation: ½×A – ½×A = ½×A + ½×E – ½×A which becomes 0 = ½×E which means 0 = E
Hint #9
eq.5 may be written as: B ÷ A = (D + E) ÷ 2 Substitute 2 for B, A for D, and 0 for E in the above equation: 2 ÷ A = (A + 0) ÷ 2 which becomes 2 ÷ A = A ÷ 2 Multiply both sides by A and 2: (2 ÷ A) × A × 2 = (A ÷ 2) × A × 2 which makes 4 = A² Since A ≠ 0 (from eq.5) and A is non-negative, the above equation makes: A = 2 and also makes D = A = 2
Hint #10
Substitute 2 for A in eq.2b: F = 2×2 + 2 which becomes F = 4 + 2 which makes F = 6
Solution
Substitute 2 for A in eq.3e: C = 2½×2 + 2 which becomes C = 5 + 2 which makes C = 7 and makes ABCDEF = 227206