Puzzle for December 22, 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 and D to both sides of eq.5: C - B + B + D = B - D + B + D which becomes eq.5a) C + D = 2×B
Hint #2
In eq.6, replace A with B + F (from eq.2), and C + D with 2×B (from eq.5a): B + F + B + F = 2×B + E 2×B + 2×F = 2×B + E Subtract 2×B from each side of the equation above: 2×B + 2×F - 2×B = 2×B + E - 2×B which makes eq.6a) 2×F = E
Hint #3
In eq.4, substitute 2×F for E: 2×F - D = D + F In the above equation, add D to both sides, and subtract F from both sides: 2×F - D + D - F = D + F + D - F which makes F = 2×D
Hint #4
Substitute (2×D) for F in eq.6a: 2×(2×D) = E which makes 4×D = E
Hint #5
Substitute 2×D for F, and 4×D for E in eq.6: A + B + 2×D = C + D + 4×D which becomes A + B + 2×D = C + 5×D Subtract 2×D from both sides of the equation above: A + B + 2×D - 2×D = C + 5×D - 2×D which becomes eq.6b) A + B = C + 3×D
Hint #6
Multiply both sides of eq.6b by 2: 2×(A + B) = 2×(C + 3×D) which becomes eq.6c) 2×A + 2×B = 2×C + 6×D
Hint #7
Substitute C + D for 2×B (from eq.5a) in eq.6c: 2×A + C + D = 2×C + 6×D Subtract C and D from each side of the equation above: 2×A + C + D - C - D = 2×C + 6×D - C - D which becomes eq.6d) 2×A = C + 5×D
Hint #8
Substitute 2×D for F in eq.3: eq.3a) C = A + 2×D
Hint #9
Substitute A + 2×D for C (from eq.3a) into eq.6d: 2×A = A + 2×D + 5×D which becomes 2×A = A + 7×D Subtract A from each side of the equation above: 2×A - A = A + 7×D - A which makes A = 7×D
Hint #10
Substitute 7×D for A, and 2×D for F in eq.2: 7×D = B + 2×D Subtract 2×D from each side of the above equation: 7×D - 2×D = B + 2×D - 2×D which makes 5×D = B
Hint #11
Substitute 7×D for A in eq.3a: C = 7×D + 2×D which makes C = 9×D
Solution
Substitute 7×D for A, 5×D for B, 9×D for C, 4×D for E, and 2×D for F in eq.1: 7×D + 5×D + 9×D + D + 4×D + 2×D = 28 which simplifies to 28×D = 28 Divide both sides of the above equation by 28: 28×D ÷ 28 = 28 ÷ 28 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 5×D = 5 × 1 = 5 C = 9×D = 9 × 1 = 9 E = 4×D = 4 × 1 = 4 F = 2×D = 2 × 1 = 2 and ABCDEF = 759142