Puzzle for January 28, 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
Subtract E from both sides of eq.3: F - E - E = E - (A + B + C + D) - E which becomes F - 2×E = -(A + B + C + D) Multiply both sides of the above equation by (-1): (-1) × (F - 2×E) = (-1) × (-(A + B + C + D)) which becomes -F + 2×E = A + B + C + D which may be written as eq.3a) 2×E - F = A + B + C + D
Hint #2
In eq.1, replace A + B + C + D with 2×E - F (from eq.3a): 2×E - F + E + F = 21 which makes 3×E = 21 Divide both sides of the above equation by 3: 3×E ÷ 3 = 21 ÷ 3 which makes E = 7
Hint #3
In eq.3a, replace E with 7: 2×7 - F = A + B + C + D which becomes 14 - F = A + B + C + D Add F to both sides of the equation above: 14 - F + F = A + B + C + D + F which becomes eq.3b) 14 = A + B + C + D + F
Hint #4
In eq.5, replace E with 7, and replace A + B + C + D + F with 14 (from eq.3b): D × 7 = 14 Divide both sides of the above equation by 7: D × 7 ÷ 7 = 14 ÷ 7 which makes D = 2
Hint #5
In eq.6, substitute 2 for D, and 7 for E: 2 × 7 = 7 + F - 2 which becomes 14 = 5 + F Subtract 5 from each side of the above equation: 14 - 5 = 5 + F - 5 which makes 9 = F
Hint #6
Substitute 2 for D, and 9 for F in eq.3b: 14 = A + B + C + 2 + 9 which becomes 14 = A + B + C + 11 Subtract 11 from both sides of the equation above: 14 - 11 = A + B + C + 11 - 11 which becomes eq.3c) 3 = A + B + C
Hint #7
Substitute 9 for F, and 7 for E in eq.2: B + C = 9 - 7 which becomes eq.2a) B + C = 2
Hint #8
Substitute 2 for B + C (from eq.2a) in eq.3c: 3 = A + 2 Subtract 2 from each side of the equation above: 3 - 2 = A + 2 - 2 which makes 1 = A
Hint #9
eq.4 may be written as: C + D = (A + B + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × (C + D) = 3 × (A + B + E) ÷ 3 which becomes eq.4a) 3×C + 3×D = A + B + E
Hint #10
Substitute 2 for D, 1 for A, and 7 for E in eq.4a: 3×C + 3×2 = 1 + B + 7 which becomes 3×C + 6 = 8 + B Subtract 8 from each side of the equation above: 3×C + 6 - 8 = 8 + B - 8 which becomes eq.4b) 3×C - 2 = B
Hint #11
Substitute 3×C - 2 for B (from eq.4b) in eq.2a: 3×C - 2 + C = 2 which becomes 4×C - 2 = 2 Add 2 to both sides of the above equation: 4×C - 2 + 2 = 2 + 2 which makes 4×C = 4 Divide both sides by 4: 4×C ÷ 4 = 4 ÷ 4 which makes C = 1
Solution
Substitute 1 for C in eq.2a: B + 1 = 2 Subtract 1 from each side of the above equation: B + 1 - 1 = 2 - 1 which makes B = 1 and makes ABCDEF = 111279