Puzzle for September 2, 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 F to each side of eq.4: E – F + F = D + F which becomes E = D + F In eq.2, replace E with D + F: D + D + F = C + F which becomes 2×D + F = C + F Subtract F from each side of the equation above: 2×D + F – F = C + F – F which makes 2×D = C
Hint #2
In eq.3, replace C with 2×D: B = 2×D + D which makes B = 3×D
Hint #3
In eq.6, substitute 2×D for C, and 3×D for B: 2×D + E = A + 3×D Subtract 2×D from both sides of the above equation: 2×D + E – 2×D = A + 3×D – 2×D which becomes eq.6a) E = A + D
Hint #4
Substitute A + D for E (from eq.6a) in eq.4: A + D – F = D In the equation above, add F to both sides, and subtract D from both sides: A + D – F + F – D = D + F – D which makes A = F
Hint #5
Substitute 3×D for B, and 2×D for C in eq.5: 3×D + 2×D = A + E + F which becomes eq.5a) 5×D = A + E + F
Hint #6
Substitute A + D for E (from eq.6a), and A for F in eq.5a: 5×D = A + A + D + A which becomes 5×D = 3×A + D Subtract D from each side of the above equation: 5×D – D = 3×A + D – D which becomes 4×D = 3×A Divide both sides by 3: 4×D ÷ 3 = 3×A ÷ 3 which makes 1⅓×D = A and also makes F = A = 1⅓×D
Hint #7
Substitute 1⅓×D for A in eq.6a: E = 1⅓×D + D which makes E = 2⅓×D
Solution
Substitute 1⅓×D for A and F, 3×D for B, 2×D for C, and 2⅓×D for E in eq.1: 1⅓×D + 3×D + 2×D + D + 2⅓×D + 1⅓×D = 33 which simplifies to 11×D = 33 Divide both sides of the equation above by 11: 11×D ÷ 11 = 33 ÷ 11 which means D = 3 making A = F = 1⅓×D = 1⅓ × 3 = 4 B = 3×D = 3 × 3 = 9 C = 2×D = 2 × 3 = 6 E = 2⅓×D = 2⅓ × 3 = 7 and ABCDEF = 496374