Puzzle for June 1, 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
Subtract C and E from each side of eq.3: A + C - C - E = B + E - C - E which becomes eq.3a) A - E = B - C Subtract E and F from each side of eq.2: C + E - E - F = A + F - E - F which becomes eq.2a) C - F = A - E
Hint #2
In eq.3a, replace A - E with C - F (from eq.2a): C - F = B - C Add F and C to both sides of the above equation: C - F + F + C = B - C + F + C which becomes eq.3b) 2×C = B + F
Hint #3
In eq.3b, replace B with C + F (from eq.1): 2×C = C + F + F which becomes 2×C = C + 2×F Subtract C from each side of the equation above: 2×C - C = C + 2×F - C which makes C = 2×F
Hint #4
In eq.1, substitute 2×F for C: B = 2×F + F which makes B = 3×F
Hint #5
Substitute 2×F for C, and 3×F for B in eq.3: A + 2×F = 3×F + E Subtract 2×F from both sides of the equation above: A + 2×F - 2×F = 3×F + E - 2×F which becomes A = F + E which may be written as eq.3c) A = E + F
Hint #6
Substitute A for E + F (from eq.3c) into eq.4: A = A + D Subtract A from both sides of the equation above: A - A = A + D - A which makes 0 = D
Hint #7
Substitute 0 for D, 3×F for B, and 2×F for C in eq.6: A - 0 = (3×F + 2×F) ÷ F which becomes A = (5×F) ÷ F which makes A = 5
Hint #8
Substitute 2×F for C, 0 for D, and 3×F for B in eq.5: 2×F + 0 = (3×F + F) ÷ E which becomes 2×F = (4×F) ÷ E Multiply both sides of the above equation by E: E × 2×F = E × (4×F) ÷ E which becomes E × 2×F = 4×F Since F ≠ 0 (from eq.6), divide both sides by 2×F: (E × 2×F) ÷ 2×F = (4×F) ÷ 2×F which makes E = 2
Solution
Substitute 5 for A, and 2 for E in eq.3c: 5 = 2 + F Subtract 2 from each side of the equation above: 5 - 2 = 2 + F - 2 which makes 3 = F making B = 3×F = 3 × 3 = 9 C = 2×F = 2 × 3 = 6 and ABCDEF = 596023