Puzzle for March 31, 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.
* AB and BC are 2-digit numbers (not A×B or B×C).
Scratchpad
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Hint #1
Add A to both sides of eq.2: E - A + A = B + D + A which becomes E = B + D + A which may be written as eq.2a) E = A + B + D
Hint #2
In eq.4, replace E with A + B + D (from eq.2a): A + C + D = B + A + B + D + F which becomes A + C + D = 2×B + A + D + F Subtract A and D from each side of the equation above: A + C + D - A - D = 2×B + A + D + F - A - D which becomes eq.4a) C = 2×B + F
Hint #3
eq.6 may be written as: 10×B + C - F = D + F In the equation above, replace C with 2×B + F (from eq.4a): 10×B + 2×B + F - F = D + F which becomes eq.6a) 12×B = D + F
Hint #4
eq.5 may be written as: 10×A + B = D + F - A Add A to both sides of the above equation: 10×A + B + A = D + F - A + A which becomes eq.5a) 11×A + B = D + F
Hint #5
In eq.5a, substitute 12×B for D + F (from eq.6a): 11×A + B = 12×B Subtract B from both sides of the equation above: 11×A + B - B = 12×B - B which makes 11×A = 11×B Divide both sides by 11: 11×A ÷ 11 = 11×B ÷ 11 which makes A = B
Hint #6
Substitute B for A in eq.4: B + C + D = B + E + F Subtract B from both sides of the above equation: B + C + D - B = B + E + F - B which becomes eq.4b) C + D = E + F
Hint #7
Subtract the left and right sides of eq.4b from the left and right sides of eq.3, respectively: D + E - (C + D) = C + F - (E + F) which becomes D + E - C - D = C + F - E - F which becomes E - C = C - E Add C and E to both sides of the equation above: E - C + C + E = C - E + C + E which makes 2×C = 2×E Divide both sides by 2: 2×C ÷ 2 = 2×E ÷ 2 which makes C = E
Hint #8
Substitute C for E in eq.4b: C + D = C + F Subtract C from each side of the above equation: C + D - C = C + F - C which makes D = F
Hint #9
Substitute D for F in eq.6a: 12×B = D + D which becomes 12×B = 2×D Divide both sides of the above equation by 2: 12×B ÷ 2 = 2×D ÷ 2 which makes 6×B = D and also makes 6×B = D = F
Hint #10
Substitute 6×B for F in eq.4a: C = 2×B + 6×B which makes C = 8×B and also makes C = E = 8×B
Solution
Substitute B for A, 8×B for C and E, and 6×B for D and F in eq.1: B + B + 8×B + 6×B + 8×B + 6×B = 30 which simplifies to 30×B = 30 Divide both sides of the above equation by 30: 30×B ÷ 30 = 30 ÷ 30 which means B = 1 making A = B = 1 C = E = 8×B = 8 × 1 = 8 D = F = 6×B = 6 × 1 = 6 and ABCDEF = 118686