Puzzle for April 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
Add B and E to both sides of eq.2: F - B + B + E = D - E + B + E which becomes F + E = D + B which is the same as eq.2a) E + F = D + B
Hint #2
In eq.4, replace E + F with D + B (from eq.2a): B + C + D = A + D + B Subtract B and D from each side of the equation above: B + C + D - B - D = A + D + B - B - D which makes C = A
Hint #3
eq.1 may be written as: A + B + C + E + D + F = 26 In the above equation, replace A + B + C + E with D (from eq.3): D + D + F = 26 which becomes 2×D + F = 26 Subtract 2×D from each side: 2×D + F - 2×D = 26 - 2×D which becomes eq.1a) F = 26 - 2×D
Hint #4
To make eq.1a true, check several possible values for D and F: If D = 9, then F = 26 - 2×D = 26 - 2×9 = 26 - 18 = 8 If D = 8, then F = 26 - 2×D = 26 - 2×8 = 26 - 16 = 10 If D < 8, then F > 10 Since F must be a one-digit non-negative integer, then: F = 8 making D = 9
Hint #5
In eq.2a, substitute 8 for F, and 9 for D: E + 8 = 9 + B Subtract 8 from each side of the equation above: E + 8 - 8 = 9 + B - 8 which becomes eq.2b) E = 1 + B
Hint #6
eq.5 may be written as: E = (B + C + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × E = 3 × (B + C + F) ÷ 3 which becomes eq.5a) 3×E = B + C + F
Hint #7
Substitute (1 + B) for E (from eq.2b), and 8 for F into eq.5a: 3×(1 + B) = B + C + 8 which becomes 3 + 3×B = B + C + 8 Subtract B and 8 from both sides of the equation above: 3 + 3×B - B - 8 = B + C + 8 - B - 8 which makes 2×B - 5 = C and also makes eq.5b) 2×B - 5 = C = A
Hint #8
Substitute 2×B - 5 for A and C (from eq.5b), 9 for D, 1 + B for E (from eq.2b), and 8 for F in eq.1: 2×B - 5 + B + 2×B - 5 + 9 + 1 + B + 8 = 26 which simplifies to 6×B + 8 = 26 Subtract 8 from each side of the equation above: 6×B + 8 - 8 = 26 - 8 which makes 6×B = 18 Divide both sides by 6: 6×B ÷ 6 = 18 ÷ 6 which makes B = 3
Solution
Since B = 3, then: A = C = 2×B - 5 = 2×3 - 5 = 6 - 5 = 1 (from eq.5b) E = 1 + B = 1 + 3 = 4 (from eq.2b) and ABCDEF = 131948