Puzzle for September 23, 2023 ( )
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 and A to both sides of eq.2: B + D - F + F + A = E - A + F + A which becomes eq.2a) B + D + A = E + F Add E and B to both sides of eq.4: A + C - E + E + B = D + F - B + E + B which becomes A + C + B = D + F + E which may be written as eq.4a) A + B + C = D + E + F
Hint #2
In eq.4a, replace E + F with B + D + A (from eq.2a): A + B + C = D + B + D + A which becomes A + B + C = 2×D + B + A Subtract A and B from each side of the equation above: A + B + C - A - B = 2×D + B + A - A - B which simplifies to eq.4b) C = 2×D
Hint #3
In eq.3, replace C with 2×D: B - D + E = 2×D + D which becomes B - D + E = 3×D Add D to both sides of the above equation: B - D + E + D = 3×D + D which becomes eq.3a) B + E = 4×D
Hint #4
In eq.5, replace C with 2×D: 2×D = average (B, F) which may be written as 2×D = (B + F) ÷ 2 Multiply both sides of the above equation by 2: 2 × 2×D = 2 × (B + F) ÷ 2 which becomes eq.5a) 4×D = B + F
Hint #5
Substitute B + F for 4×D (from eq.5a) into eq.3a: B + E = B + F Subtract B from each side of the above equation: B + E - B = B + F - B which makes E = F
Hint #6
Substitute E for F in eq.2a: B + D + A = E + E which becomes eq.2b) B + D + A = 2×E
Hint #7
eq.1 may be written as: B + D + A + C + E + F = 44 Substitute 2×E for B + D + A (from eq.2b), and E for F in the equation above: 2×E + C + E + E = 44 which becomes 4×E + C = 44 Subtract 4×E from both sides: 4×E + C - 4×E = 44 - 4×E which makes eq.1a) C = 44 - 4×E
Hint #8
To make eq.1a true, check several possible values for E and C: If E = 9, then C = 44 - 4×9 = 44 - 36 = 8 If E = 8, then C = 44 - 4×8 = 44 - 32 = 12 If E < 8, then C > 12 Since C must be a one-digit non-negative integer, then from the above equations: C = 8 which makes E = 9 and also makes E = F = 9
Hint #9
Substitute 8 for C in eq.4b: 8 = 2×D Divide both sides of the above equation by 2: 8 ÷ 2 = 2×D ÷ 2 which makes 4 = D
Hint #10
Substitute 9 for E, and 4 for D in eq.3a: B + 9 = 4×4 which becomes B + 9 = 16 Subtract 9 from both sides of the equation above: B + 9 - 9 = 16 - 9 which makes B = 7
Solution
Substitute 7 for B, 4 for D, and 9 for E and F in eq.2a: 7 + 4 + A = 9 + 9 which becomes 11 + A = 18 Subtract 11 from each side of the equation above: 11 + A - 11 = 18 - 11 which makes A = 7 and makes ABCDEF = 778499