Puzzle for November 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 D and C to both sides of eq.2: C - D + D + C = B - C + D + C which becomes eq.2a) 2×C = B + D Add F and A to both sides of eq.4: E - F + F + A = F - A + F + A which becomes eq.4a) E + A = 2×F
Hint #2
eq.6 may be written as: F = (A + B + D + E) ÷ 4 Multiply both sides of the above equation by 4: 4 × F = 4 × (A + B + D + E) ÷ 4 which becomes eq.6a) 4×F = A + B + D + E which may be written as eq.6b) 4×F = B + D + E + A
Hint #3
In eq.6b, replace B + D with 2×C (from eq.2a), and E + A with 2×F (from eq.4a): 4×F = 2×C + 2×F Subtract 2×F from each side of the equation above: 4×F - 2×F = 2×C + 2×F - 2×F which makes 2×F = 2×C Divide both sides by 2: 2×F ÷ 2 = 2×C ÷ 2 which makes F = C
Hint #4
In eq.6a, replace F with C, and A + B + D with C + E (from eq.5): 4×C = C + E + E which becomes 4×C = C + 2×E Subtract C from each side of the above equation: 4×C - C = C + 2×E - C which makes 3×C = 2×E Divide both sides by 2: 3×C ÷ 2 = 2×E ÷ 2 which makes 1½×C = E
Hint #5
In eq.4a, substitute 1½×C for E, and C for F: 1½×C + A = 2×C Subtract 1½×C from both sides of the above equation: 1½×C + A - 1½×C = 2×C - 1½×C which makes A = ½×C
Hint #6
Substitute 1½×C for E in eq.3: D - 1½×C = B - D Add D to both sides of the equation above: D - 1½×C + D = B - D + D which becomes eq.3a) 2×D - 1½×C = B
Hint #7
Substitute 2×D - 1½×C for B (from eq.3a) in eq.2: C - D = 2×D - 1½×C - C which becomes C - D = 2×D - 2½×C Add D and 2½×C to both sides of the above equation: C - D + D + 2½×C = 2×D - 2½×C + D + 2½×C which makes 3½×C = 3×D Divide both sides by 3: 3½×C ÷ 3 = 3×D ÷ 3 which makes 1⅙×C = D
Hint #8
Substitute (1⅙×C) for D in eq.3a: 2×(1⅙×C) - 1½×C = B which becomes 2⅓×C - 1½×C = B which makes ⅚×C = B
Solution
Substitute ½×C for A, ⅚×C for B, 1⅙×C for D, 1½×C for E, and C for F in eq.1: ½×C + ⅚×C + C + 1⅙×C + 1½×C + C = 36 which simplifies to 6×C = 36 Divide both sides of the above equation by 6: 6×C ÷ 6 = 36 ÷ 6 which means C = 6 making A = ½×C = ½ × 6 = 3 B = ⅚×C = ⅚ × 6 = 5 D = 1⅙×C = 1⅙ × 6 = 7 E = 1½×C = 1½ × 6 = 9 F = C = 6 and ABCDEF = 356796