Puzzle for August 16, 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.
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Hint #1
eq.6 may be written as: A = (B + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × (B + C + E) ÷ 3 which becomes eq.6a) 3×A = B + C + E Add F to both sides of eq.4: A + F + F = C + E - F + F which becomes eq.4a) A + 2×F = C + E
Hint #2
In eq.6a, replace C + E with A + 2×F (from eq.4a): 3×A = B + A + 2×F Subtract A from each side of the equation above: 3×A - A = B + A + 2×F - A which becomes 2×A = B + 2×F Divide both sides by 2: 2×A ÷ 2 = (B + 2×F) ÷ 2 which becomes eq.6b) A = ½×B + F
Hint #3
In eq.3, replace A with ½×B + F (from eq.6b): E = ½×B + F + B which becomes eq.3a) E = 1½×B + F
Hint #4
In eq.5, substitute 1½×B + F for E (from eq.3a), and B + F for D (from eq.2): B + 1½×B + F = B + F + F - B which becomes 2½×B + F = 2×F Subtract F from both sides of the above equation: 2½×B + F - F = 2×F - F which becomes 2½×B = F
Hint #5
Substitute 2½×B for F in eq.3a: E = 1½×B + 2½×B which makes E = 4×B
Hint #6
Substitute 2½×B for F in eq.6b: A = ½×B + 2½×B which makes A = 3×B
Hint #7
Substitute 2½×B for F in eq.2: D = B + 2½×B which makes D = 3½×B
Hint #8
Substitute (3×B) for A, and 4×B for E in eq.6a: 3×(3×B) = B + C + 4×B which becomes 9×B = 5×B + C Subtract 5×B from both sides of the above equation: 9×B - 5×B = 5×B + C - 5×B which makes 4×B = C
Solution
Substitute 3×B for A, 4×B for C and E, 3½×B for D, and 2½×B for F in eq.1: 3×B + B + 4×B + 3½×B + 4×B + 2½×B = 36 which simplifies to 18×B = 36 Divide both sides of the above equation by 18: 18×B ÷ 18 = 36 ÷ 18 which means B = 2 making A = 3×B = 3 × 2 = 6 C = E = 4×B = 4 × 2 = 8 D = 3½×B = 3½ × 2 = 7 F = 2½×B = 2½ × 2 = 5 and ABCDEF = 628785