Puzzle for November 26, 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 positive integer.
Scratchpad
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Hint #1
In eq.5, replace D + E with B + F (from eq.2): B + F – A = A + F In the equation above, subtract F from both sides, and add A to both sides: B + F – A – F + A = A + F – F + A which makes B = 2×A
Hint #2
Add C and F to both sides of eq.3: E – C + C + F = C – F + C + F which becomes eq.3a) E + F = 2×C In eq.6, replace E + F with 2×C (from eq.3a): (C × D) + 2×C = B × C which may be written as C × (D + 2) = B × C Divide both sides of the equation above by C: C × (D + 2) ÷ C = B × C ÷ C which means eq.6a) D + 2 = B
Hint #3
In eq.6a, substitute 2×A for B: D + 2 = 2×A Subtract 2 from each side of the above equation: D + 2 – 2 = 2×A – 2 which becomes eq.6b) D = 2×A – 2
Hint #4
Substitute 2×A for B, and 2×A – 2 for D (from eq.6b) in eq.4: 2×A – E = A – 2×A + 2×A – 2 which becomes 2×A – E = A – 2 In the equation above, add E and 2 to both sides, and subtract A from both sides: 2×A – E + E + 2 – A = A – 2 + E + 2 – A which makes eq.4a) A + 2 = E
Hint #5
Substitute A + 2 for E (from eq.4a), and D + 2 for B (from eq.6a) in eq.2: D + A + 2 = D + 2 + F Subtract D and 2 from both sides of the equation above: D + A + 2 – D – 2 = D + 2 + F – D – 2 which makes A = F
Hint #6
Substitute A + 2 for E (from eq.4a), and A for F in eq.3: A + 2 – C = C – A Add C and A to each side of the above equation: A + 2 – C + C + A = C – A + C + A which becomes 2×A + 2 = 2×C Divide each side by 2: (2×A + 2) ÷ 2 = 2×C ÷ 2 which makes eq.3a) A + 1 = C
Solution
Substitute 2×A for B, A + 1 for C (from eq.3a), 2×A – 2 for D (from eq.6b), A + 2 for E (from eq.4a), and A for F in eq.1: A + 2×A + A + 1 + 2×A – 2 + A + 2 + A = 33 which simplifies to 8×A + 1 = 33 Subtract 1 from both sides of the above equation: 8×A + 1 – 1 = 33 – 1 which makes 8×A = 32 Divide both sides by 8: 8×A ÷ 8 = 32 ÷ 8 which means A = 4 making B = 2×A = 2 × 4 = 8 C = A + 1 = 4 + 1 = 5 (from eq.3a) D = 2×A – 2 = 2×4 – 2 = 8 – 2 = 6 (from eq.6b) E = A + 2 = 4 + 2 = 6 (from eq.4a) F = A = 4 and ABCDEF = 485664