Puzzle for March 28, 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 to both sides of eq.4: B - D + B = F - B + B which becomes 2×B - D = F In eq.2, replace F with 2×B - D: E = D + 2×B - D which makes eq.2a) E = 2×B
Hint #2
In eq.6, replace E with 2×B: 2×B - A = A + B In the above equation, add A to both sides, and subtract B from both sides: 2×B - A + A - B = A + B + A - B which simplifies to B = 2×A
Hint #3
In eq.2a, substitute (2×A) for B: E = 2×(2×A) which makes E = 4×A
Hint #4
Substitute 2×A for B in eq.5: F - 2×A = A - F Add 2×A and F to both sides of the above equation: F - 2×A + 2×A + F = A - F + 2×A + F which simplifies to 2×F = 3×A Divide both sides by 2: 2×F ÷ 2 = 3×A ÷ 2 which makes F = 1½×A
Hint #5
Substitute 2×A for B, and 1½×A for F in eq.4: 2×A - D = 1½×A - 2×A which becomes 2×A - D = -½×A Add D and ½×A to both sides of the equation above: 2×A - D + D + ½×A = -½×A + D + ½×A which makes 2½×A = D
Hint #6
Substitute 2½×A for D in eq.3: C - 2½×A = 2½×A - A which becomes C - 2½×A = 1½×A Add 2½×A to both sides of the equation above: C - 2½×A + 2½×A = 1½×A + 2½×A which makes C = 4×A
Solution
Substitute 2×A for B, 4×A for C and E, 2½×A for D, and 1½×A for F in eq.1: A + 2×A + 4×A + 2½×A + 4×A + 1½×A = 30 which simplifies to 15×A = 30 Divide both sides of the above equation by 15: 15×A ÷ 15 = 30 ÷ 15 which means A = 2 making B = 2×A = 2 × 2 = 4 C = E = 4×A = 4 × 2 = 8 D = 2½×A = 2½ × 2 = 5 F = 1½×A = 1½ × 2 = 3 and ABCDEF = 248583