Puzzle for April 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
In eq.3, replace E with A + C (from eq.4): A + C + F – A = C + D which becomes C + F = C + D Subtract C from each side of the equation above: C + F – C = C + D – C which makes F = D
Hint #2
In eq.2, replace D with F, and replace A + B with E + F – C (from eq.6): C + F – A = E + F – C In the equation above, subtract F from both sides, and add C to both sides: C + F – A – F + C = E + F – C – F + C which simplifies to eq.2a) 2×C – A = E
Hint #3
In eq.4, substitute 2×C – A for E (from eq.2a): 2×C – A = A + C In the equation above, add A to both sides, and subtract C from each side: 2×C – A + A – C = A + C + A – C which makes C = 2×A
Hint #4
Substitute (2×A) for C in eq.2a: 2×(2×A) – A = E which becomes 4×A – A = E which makes 3×A = E
Hint #5
Substitute 3×A for E, D for F, and 2×A for C in eq.5: D – 3×A + D = A + 2×A + 3×A which becomes 2×D – 3×A = 6×A Add 3×A to each side of the above equation: 2×D – 3×A + 3×A = 6×A + 3×A which makes 2×D = 9×A Divide both sides by 2: 2×D ÷ 2 = 9×A ÷ 2 which makes D = 4½×A and also makes F = D = 4½×A
Hint #6
Substitute 2×A for C, and 4½×A for D in eq.2: 2×A + 4½×A – A = A + B which becomes 5½×A = A + B Subtract A from each side of the equation above: 5½×A – A = A + B – A which makes 4½×A = B
Solution
Substitute 4½×A for B and D and F, 2×A for C, and 3×A for E in eq.1: A + 4½×A + 2×A + 4½×A + 3×A + 4½×A = 39 which simplifies to 19½×A = 39 Divide both sides of the equation above by 19½: 19½×A ÷ 19½ = 39 ÷ 19½ which means A = 2 making B = D = F = 4½×A = 4½ × 2 = 9 C = 2×A = 2 × 2 = 4 E = 3×A = 3 × 2 = 6 and ABCDEF = 294969