Puzzle for September 9, 2022 ( )
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.3 may be written as: A - F = (B + E) ÷ 2 Multiply both sides of the above equation by 2: 2 × (A - F) = 2 × (B + E) ÷ 2 which becomes eq.3a) 2×A - 2×F = B + E
Hint #2
In eq.2, replace B + E with 2×A - 2×F (from eq.3a): 2×A - 2×F - F = A + F which becomes 2×A - 3×F = A + F In the equation above, add 3×F to both sides, and subtract A from both sides: 2×A - 3×F + 3×F - A = A + F + 3×F - A which simplifies to eq.2a) A = 4×F
Hint #3
eq.4 may be written as: A = (C + D + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (C + D + E) ÷ 3 which becomes eq.4a) 3×A = C + D + E
Hint #4
In eq.4a, replace A with 4×F (from eq.2a): 3×(4×F) = C + D + E which becomes eq.4b) 12×F = C + D + E
Hint #5
In eq.1, substitute 4×F for A (from eq.2a), and 12×F for C + D + E (from eq.4b): 4×F + B + 12×F + F = 17 which becomes 17×F + B = 17 Subtract 17×F from each side of the equation above: 17×F + B - 17×F = 17 - 17×F which becomes eq.1a) B = 17 - 17×F
Hint #6
To make eq.1a true, check several possible values for F and B: If F = 0, then B = 17 - 17×0 = 17 - 0 = 17 If F = 1, then B = 17 - 17×1 = 17 - 17 = 0 If F = 2, then B = 17 - 17×2 = 17 - 34 = -17 If F > 2, then B < -17 Since B must be a non-negative one-digit integer, the above equations make: F = 1 and B = 0
Hint #7
Substitute 1 for F in eq.2a: A = 4×1 which makes A = 4
Hint #8
Substitute 4 for A, 1 for F, and 0 for B in eq.3a: 2×4 - 2×1 = 0 + E which becomes 8 - 2 = E which makes 6 = E
Hint #9
Substitute 1 for F, and 6 for E in eq.4b: 12×1 = C + D + 6 which becomes 12 = C + D + 6 Subtract 6 from each side of the equation above: 12 - 6 = C + D + 6 - 6 which becomes eq.4c) 6 = C + D
Hint #10
Substitute 1 for F, and 6 for E in eq.5: C + 1 = D ÷ 6 Multiply both sides of the above equation by 6: 6 × (C + 1) = 6 × (D ÷ 6) which becomes eq.5a) 6×C + 6 = D
Hint #11
In eq.5a, substitute C + D for 6 (from eq.4c): 6×C + C + D = D which becomes 7×C + D = D Subtract D from each side of the equation above: 7×C + D - D = D - D which makes 7×C = 0 which means C = 0
Solution
Substitute 0 for C in eq.4c: 6 = 0 + D which makes 6 = D and makes ABCDEF = 400661