Puzzle for November 19, 2022 ( )
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 F and D to both sides of eq.3: E - F + F + D = F - D + F + D which becomes eq.3a) E + D = 2×F Add B and F to both sides of eq.4: F - B + B + F = A - D - F + B + F which becomes eq.4a) 2×F = A - D + B
Hint #2
In eq.4a, replace 2×F with E + D (from eq.3a): E + D = A - D + B Add D to both sides of the above equation: E + D + D = A - D + B + D which becomes eq.4b) E + 2×D = A + B
Hint #3
Add D and A to both sides of eq.6: F - D + D + A = D - A + D + A which becomes eq.6a) F + A = 2×D
Hint #4
In eq.4b, replace 2×D with F + A (from eq.6a): E + F + A = A + B Subtract A from each side of the equation above: E + F + A - A = A + B - A which becomes eq.4c) E + F = B
Hint #5
In eq.2, substitute E + F for B (from eq.4c): E + F = C + F Subtract F from each side of the equation above: E + F - F = C + F - F which makes E = C
Hint #6
Substitute C for E in eq.3a: eq.3b) C + D = 2×F Add D and F to both sides of eq.5: D + F + D + F = B + C - D - F + D + F which becomes eq.5a) 2×D + 2×F = B + C
Hint #7
Substitute C + D for 2×F (from eq.3b) in eq.5a: 2×D + C + D = B + C which becomes 3×D + C = B + C Subtract C from each side of the above equation: 3×D + C - C = B + C - C which becomes eq.5b) 3×D = B
Hint #8
Substitute 3×D for B in eq.4a: 2×F = A - D + 3×D which becomes eq.4d) 2×F = A + 2×D
Hint #9
Substitute F + A for 2×D (from eq.6a) in eq.4d: 2×F = A + F + A which becomes 2×F = 2×A + F Subtract F from both sides of the above equation: 2×F - F = 2×A + F - F which makes F = 2×A
Hint #10
Substitute 2×A for F in eq.6a: 2×A + A = 2×D which makes 3×A = 2×D Divide both sides of the above equation by 2: 3×A ÷ 2 = 2×D ÷ 2 which makes 1½×A = D
Hint #11
Substitute (1½×A) for D in eq.5b: 3×(1½×A) = B which makes 4½×A = B
Hint #12
Substitute 1½×A for D, and (2×A) for F in eq.3b: C + 1½×A = 2×(2×A) which becomes C + 1½×A = 4×A Subtract 1½×A from each side of the equation above: C + 1½×A - 1½×A = 4×A - 1½×A which makes C = 2½×A and also makes E = C = 2½×A
Solution
Substitute 4½×A for B, 2½×A for C and E, 1½×A for D, and 2×A for F in eq.1: A + 4½×A + 2½×A + 1½×A + 2½×A + 2×A = 28 which simplifies to 14×A = 28 Divide both sides of the above equation by 14: 14×A ÷ 14 = 28 ÷ 14 which means A = 2 making B = 4½×A = 4½ × 2 = 9 C = E = 2½×A = 2½ × 2 = 5 D = 1½×A = 1½ × 2 = 3 F = 2×A = 2 × 2 = 4 and ABCDEF = 295354