Puzzle for September 5, 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: C + B = A - B + B which becomes eq.3a) C + B = A
Hint #2
In eq.5, substitute (B - C) for F (from eq.2), and C + B for A (from eq.3a): (B - C) - C = C + B - (B - C) which becomes B - 2×C = C + B - B + C which becomes B - 2×C = 2×C Add 2×C to both sides of the above equation: B - 2×C + 2×C = 2×C + 2×C which makes B = 4×C
Hint #3
In eq.2, replace B with 4×C: F = 4×C - C which makes F = 3×C
Hint #4
In eq.3, replace B with 4×C: C = A - 4×C Add 4×C to both sides of the equation above: C + 4×C = A - 4×C + 4×C which makes 5×C = A
Hint #5
Substitute 4×C for B, and 3×C for F in eq.4: D - 4×C = 4×C - 3×C which becomes D - 4×C = C Add 4×C to both sides of the above equation: D - 4×C + 4×C = C + 4×C which makes D = 5×C
Hint #6
Substitute 5×C for A and D in eq.6: C - E = 5×C ÷ 5×C which becomes C - E = 1 In the above equation, add E to both sides, and subtract 1 from both sides: C - E + E - 1 = 1 + E - 1 which makes eq.6a) C - 1 = E
Hint #7
Substitute 5×C for A and D, 4×C for B, C - 1 for E (from eq.6a), and 3×C for F in eq.1: 5×C + 4×C + C + 5×C + C - 1 + 3×C = 18 which simplifies to 19×C - 1 = 18 Add 1 to both sides of the equation above: 19×C - 1 + 1 = 18 + 1 which makes 19×C = 19 Divide both sides by 19: 19×C ÷ 19 = 19 ÷ 19 which means C = 1
Solution
Since C = 1, then: A = D = 5×C = 5×1 = 5 B = 4×C = 4×1 = 4 E = C - 1 = 1 - 1 = 0 (from eq.6a) F = 3×C = 3×1 = 3 and ABCDEF = 541503