Puzzle for December 4, 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
In eq.4, replace A with D + E (from eq.1): D - E = D + E - D which becomes D - E = E Add E to both sides of the above equation: D - E + E = E + E which makes D = 2×E
Hint #2
In eq.1, replace D with 2×E: A = 2×E + E which makes A = 3×E
Hint #3
Add D to both sides of eq.3: C - D + D = D - F + D which becomes C = 2×D - F In eq.2, replace C with 2×D - F: B = 2×D - F + F which makes eq.2a) B = 2×D
Hint #4
In eq.2a, substitute (2×E) for D: B = 2×(2×E) which makes B = 4×E
Hint #5
Substitute 4×E for B, and 2×E for D in eq.5: E = 4×E ÷ 2×E which makes E = 2 and also makes A = 3×E = 3 × 2 = 6 B = 4×E = 4 × 2 = 8 D = 2×E = 2 × 2 = 4
Hint #6
Substitute 2 for E, 4 for D, and 8 for B in eq.6: F = (2 ÷ 4) + (4 ÷ 8) which becomes F = ½ + ½ which makes F = 1
Solution
Substitute 8 for B, and 1 for F in eq.2: 8 = C + 1 Subtract 1 from each side of the equation above: 8 - 1 = C + 1 - 1 which makes 7 = C and makes ABCDEF = 687421