Puzzle for September 29, 2022 ( )
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.2: D + B = E - B + B which becomes D + B = E which may be written as B + D = E In eq.3, replace B + D with E: eq.3a) E = C - D
Hint #2
In eq.4, replace E with C - D (from eq.3a): C + D = C - D - D which becomes C + D = C - 2×D In the above equation, subtract C from both sides, and add 3×D to both sides: C + D - C + 2×D = C - 2×D - C + 2×D which becomes 3×D = 0 which means D = 0
Hint #3
In eq.3a, substitute 0 for D: E = C - 0 which makes E = C
Hint #4
In eq.3, substitute 0 for D: B + 0 = C - 0 which makes B = C
Hint #5
Substitute C for E and B in eq.5: A + C = C + C + F which becomes A + C = 2×C + F Subtract C from each side of the equation above: A + C - C = 2×C + F - C which becomes eq.5a) A = C + F
Hint #6
Substitute C + F for A (from eq.5a), and C for B and E in eq.6: C + F + C - F = (C ÷ C) + F which becomes 2×C = 1 + F Subtract 1 from each side of the above equation: 2×C - 1 = 1 + F - 1 which becomes eq.6a) 2×C - 1 = F
Hint #7
Substitute 2×C - 1 for F (from eq.6a) in eq.5a: A = C + 2×C - 1 which becomes eq.5b) A = 3×C - 1
Hint #8
Substitute 3×C - 1 for A (from eq.5b), C for B and E, 0 for D, and 2×C - 1 for F (from eq.6a) in eq.1: 3×C - 1 + C + C + 0 + C + 2×C - 1 = 22 which simplifies to 8×C - 2 = 22 Add 2 to both sides of the equation above: 8×C - 2 + 2 = 22 + 2 which makes 8×C = 24 Divide both sides by 8: 8×C ÷ 8 = 24 ÷ 8 which means C = 3
Solution
Since C = 3, then: A = 3×C - 1 = 3×3 - 1 = 9 - 1 = 8 (from eq.5b) B = E = C = 3 F = 2×C - 1 = 2×3 - 1 = 6 - 1 = 5 (from eq.6a) and ABCDEF = 833035