Puzzle for November 13, 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
eq.4 may be written as: A + F + B = D + E In the above equation, replace A + F with D (from eq.2): D + B = D + E Subtract D from both sides: D + B – D = D + E – D which makes B = E
Hint #2
In eq.5, replace E with B, and D with A + F (from eq.2): B + B – A = A + A + F – F which becomes 2×B – A = 2×A Add A to both sides of the above equation: 2×B – A + A = 2×A + A which becomes 2×B = 3×A Divide both sides by 2: 2×B ÷ 2 = 3×A ÷ 2 which makes B = 1½×A and also makes E = B = 1½×A
Hint #3
eq.6 may be written as: D + F = (A + B + C + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × (D + F) = 4 × (A + B + C + E) ÷ 4 which becomes eq.6a) 4×D + 4×F = A + B + C + E
Hint #4
eq.1 may be written as: A + B + C + E + D + F = 40 Substitute 4×D + 4×F for A + B + C + E (from eq.6a) in the above equation: 4×D + 4×F + D + F = 40 which becomes 5×D + 5×F = 40 Divide both sides by 5: (5×D + 5×F) ÷ 5 = 40 ÷ 5 which becomes D + F = 8 Subtract F from each side: D + F – F = 8 – F which becomes eq.1a) D = 8 – F
Hint #5
Substitute 8 – F for D (from eq.1a) in eq.2: 8 – F = A + F In the above equation, add F to each side, and subtract A from each side: 8 – F + F – A = A + F + F – A which becomes 8 – A = 2×F Divide both sides by 2: (8 – A) ÷ 2 = 2×F ÷ 2 which makes eq.2a) 4 – ½×A = F
Hint #6
Substitute (4 – ½×A) for F (from eq.2a) in eq.1a: D = 8 – (4 – ½×A) which is equivalent to D = 8 – 4 + ½×A which makes eq.1b) D = 4 + ½×A
Hint #7
Substitute 4 + ½×A for D (from eq.1b), and 1½×A for E in eq.3: C + 4 + ½×A = A + 1½×A which becomes C + 4 + ½×A = 2½×A Subtract 4 and ½×A from both sides of the equation above: C + 4 + ½×A – 4 – ½×A = 2½×A – 4 – ½×A which makes eq.3a) C = 2×A – 4
Solution
Substitute 1½×A for B and E, 2×A – 4 for C (from eq.3a), 4 + ½×A for D (from eq.1b), and 4 – ½×A for F (from eq.2a) in eq.1: A + 1½×A + 2×A – 4 + 4 + ½×A + 1½×A + 4 – ½×A = 40 which simplifies to 6×A + 4 = 40 Subtract 4 from both sides of the equation above: 6×A + 4 – 4 = 40 – 4 which makes 6×A = 36 Divide both sides by 6: 6×A ÷ 6 = 36 ÷ 6 which means A = 6 making B = E = 1½×A = 1½ × 6 = 9 C = 2×A – 4 = 2×6 – 4 = 12 – 4 = 8 (from eq.3a) D = 4 + ½×A = 4 + ½×6 = 4 + 3 = 7 (from eq.1b) F = 4 – ½×A = 4 – ½×6 = 4 – 3 = 1 (from eq.2a) and ABCDEF = 698791