Puzzle for February 14, 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
Add B and D to both sides of eq.4: D – B + B + D = A – D + B + D which becomes eq.4a) 2×D = A + B Replace A + B with 2×D (from eq.4a) in eq.2: E = 2×D + D which makes E = 3×D
Hint #2
eq.3 may be written as: C – (A + B + D) = E – F In the above equation, replace (A + B + D) with E (from eq.2): C – E = E – F Add E and F to each side: C – E + E + F = E – F + E + F which becomes eq.3a) C + F = 2×E
Hint #3
In eq.3a, substitute (3×D) for E: C + F = 2×(3×D) which becomes eq.3b) C + F = 6×D
Hint #4
eq.1 may be written as: A + B + D + C + F + E = 36 Substitute 2×D for A + B (from eq.4a), 6×D for C + F (from eq.3b), and 3×D for E in the above equation: 2×D + D + 6×D + 3×D = 36 which becomes 12×D = 36 Divide both sides of the equation above by 12: 12×D ÷ 12 = 36 ÷ 12 which makes D = 3 making E = 3×D = 3×3 = 9
Hint #5
Substitute 9 for E in eq.3a: C + F = 2×9 which makes eq.3b) C + F = 18 To make eq.3b true, check several possible values for C and F: If C = 9, then 9 + F = 18 which means F = 9 If C = 8, then 8 + F = 18 which means F = 10 If C < 8, then F > 10 Since C and F must be one-digit integers, this means C = 9 which makes F = 9
Hint #6
Add B to both sides of eq.5: B + E + B = A – B + C + B which becomes 2×B + E = A + C Substitute 9 for C and E in the equation above: 2×B + 9 = A + 9 Subtract 9 from both sides: 2×B + 9 – 9 = A + 9 – 9 which makes 2×B = A
Solution
Substitute 3 for D, and 2×B for A in eq.4a: 2×3 = 2×B + B which makes 6 = 3×B Divide both sides of the equation above by 3: 6 ÷ 3 = 3×B ÷ 3 which makes 2 = B making A = 2×B = 2 × 2 = 4 and ABCDEF = 429399