Puzzle for June 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 B to both sides of eq.3: B + C + B = A - B + D + B which becomes eq.3a) 2×B + C = A + D
Hint #2
In eq.3a, replace A + D with C + F (from eq.2): 2×B + C = C + F Subtract C from each side of the equation above: 2×B + C - C = C + F - C which becomes 2×B = F
Hint #3
eq.5 may be written as: D + E = B + C + F - D In the above equation, replace C + F with A + D (from eq.2): D + E = B + A + D - D which becomes eq.5a) D + E = B + A
Hint #4
In eq.4, replace F with 2×B: B + E = A + D + 2×B Subtract B from each side of the above equation: B + E - B = A + D + 2×B - B which becomes E = A + D + B which may be written as eq.4a) E = B + A + D
Hint #5
In eq.4a, substitute D + E for B + A (from eq.5a): E = D + E + D which becomes E = 2×D + E Subtract E from both sides of the above equation: E - E = 2×D + E - E which simplifies to 0 = 2×D which means 0 = D
Hint #6
Substitute 0 for D in eq.3a: 2×B + C = A + 0 which becomes eq.3b) 2×B + C = A
Hint #7
Substitute 0 for D, and 2×B + C for A (from eq.3b) in eq.5a: 0 + E = B + 2×B + C which becomes eq.5b) E = 3×B + C
Hint #8
Substitute 2×B + C for A (from eq.3b), 2×B for F, and 3×B + C for E (from eq.5b) in eq.6: 2×B + C + C - 2×B = B + 3×B + C + 2×B which becomes 2×C = 6×B + C Subtract C from both sides of the equation above: 2×C - C = 6×B + C - C which makes C = 6×B
Hint #9
Substitute 6×B for C in eq.3b: 2×B + 6×B = A which makes 8×B = A
Hint #10
Substitute 6×B for C in eq.5b: E = 3×B + 6×B which makes E = 9×B
Solution
Substitute 8×B for A, 6×B for C, 0 for D, 9×B for E, and 2×B for F in eq.1: 8×B + B + 6×B + 0 + 9×B + 2×B = 26 which simplifies to 26×B = 26 Divide both sides of the above equation by 26: 26×B ÷ 26 = 26 ÷ 26 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 6×B = 6 × 1 = 6 E = 9×B = 9 × 1 = 9 F = 2×B = 2 × 1 = 2 and ABCDEF = 816092