Puzzle for December 17, 2019 ( )
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.4, replace D with A + C (from eq.2): C + A + C = A + F which becomes A + 2×C = A + F Subtract A from both sides of the above equation: A + 2×C – A = A + F – A which makes 2×C = F
Hint #2
In eq.6, replace F with 2×C: 2×C – A = C In the above equation, add A to both sides, and subtract C from each side: 2×C – A + A – C = C + A – C which makes C = A
Hint #3
In eq.4, substitute C for A, and 2×C for F: C + D = C + 2×C which becomes C + D = 3×C Subtract C from each side of the equation above: C + D – C = 3×C – C which makes D = 2×C and means eq.4a) F = D = 2×C
Hint #4
Substitute C for A, and 2×C for F in eq.5: E – B = C – C – 2×C which becomes E – B = –2×C Add B and 2×C to both sides of the equation above: E – B + B + 2×C = –2×C + B + 2×C which becomes eq.5a) E + 2×C = B
Hint #5
Substitute E + 2×C for B (from eq.5a), and 2×C for D in eq.3: E + 2×C + E = C + 2×C – E which becomes 2×C + 2×E = 3×C – E In the above equation, add E to both sides, and subtract 2×C from each side: 2×C + 2×E + E – 2×C = 3×C – E + E – 2×C which makes 3×E = C and also makes A = C = 3×E
Hint #6
Substitute (3×E) for C in eq.4a: F = D = 2×(3×E) which makes F = D = 6×E
Hint #7
Substitute (3×E) for C in eq.5a: E + 2×(3×E) = B which becomes E + 6×E = B which makes 7×E = B
Solution
Substitute 3×E for A and C, 7×E for B, and 6×E for D and F in eq.1: 3×E + 7×E + 3×E + 6×E + E + 6×E = 26 which simplifies to 26×E = 26 Divide both sides of the equation above by 26: 26×E ÷ 26 = 26 ÷ 26 which means E = 1 making A = C = 3×E = 3 × 1 = 3 B = 7×E = 7 × 1 = 7 D = F = 6×E = 6 × 1 = 6 and ABCDEF = 373616