Puzzle for October 21, 2021 ( )
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
In eq.4, replace F with B + D (from eq.2): E + B + D – B = A + D which becomes E + D = A + D In the above equation, subtract D from both sides: E + D – D = A + D – D which makes E = A
Hint #2
In eq.5, replace F with B + D (from eq.2): D – E + B + D = B + E which becomes 2×D – E + B = B + E In the equation above, add E to both sides, and subtract B from both sides: 2×D – E + B + E – B = B + E + E – B which simplifies to 2×D = 2×E Divide both sides by 2: 2×D ÷ 2 = 2×E ÷ 2 which makes D = E
Hint #3
eq.6 may be written as: C = (B + D + F) ÷ 3 Multiply both sides of the equation above by 3: 3 × C = 3 × (B + D + F) ÷ 3 which becomes eq.6a) 3×C = B + D + F
Hint #4
In eq.6a, replace B + D with F (from eq.2): 3×C = F + F which makes 3×C = 2×F Divide both sides of the above equation by 2: 3×C ÷ 2 = 2×F ÷ 2 which makes eq.6b) 1½×C = F
Hint #5
In eq.3, replace A with E: B + C = E – B Add B to both sides of the equation above: B + C + B = E – B + B which makes 2×B + C = E and also makes eq.3a) A = E = D = 2×B + C
Hint #6
Substitute 2×B + C for D (from eq.3a), and 1½×C for F in eq.6a: 3×C = B + 2×B + C + 1½×C which becomes 3×C = 3×B + 2½×C Subtract 2½×C from each side of the equation above: 3×C – 2½×C = 3×B + 2½×C – 2½×C which makes ½×C = 3×B Multiply both sides by 2: 2 × (½×C) = 2 × (3×B) which makes C = 6×B
Hint #7
Substitute 6×B for C in eq.3a: A = E = D = 2×B + 6×B which makes A = E = D = 8×B
Hint #8
Substitute (6×B) for C in eq.6b: 1½×(6×B) = F which makes 9×B = F
Solution
Substitute 8×B for A and D and E, 6×B for C, and 9×B for F in eq.1: 8×B + B + 6×B + 8×B + 8×B + 9×B = 40 which simplifies to 40×B = 40 Divide both sides of the equation above by 40: 40×B ÷ 40 = 40 ÷ 40 which means B = 1 making A = D = E = 8×B = 8 × 1 = 8 C = 6×B = 6 × 1 = 6 F = 9×B = 9 × 1 = 9 and ABCDEF = 816889