Puzzle for December 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
In eq.4, replace C with A + B (from eq.2): B - A = A + B - B which becomes B - A = A Add A to both sides of the above equation: B - A + A = A + A which makes B = 2×A
Hint #2
In eq.2, replace B with 2×A: C = A + 2×A which makes C = 3×A
Hint #3
In eq.6, substitute 2×A for B, and 3×A for C: D = A + 2×A + 3×A which makes D = 6×A
Hint #4
Substitute E + F for D (from eq.3) in eq.5: E - F = E + F - E which becomes E - F = F Add F to both sides of the equation above: E - F + F = F + F which makes eq.5a) E = 2×F
Hint #5
Substitute 6×A for D, and 2×F for E in eq.3: 6×A = 2×F + F which makes 6×A = 3×F Divide both sides of the above equation by 3: 6×A ÷ 3 = 3×F ÷ 3 which makes 2×A = F
Hint #6
Substitute 2×A for F in eq.5a: E = 2×(2×A) which makes E = 4×A
Solution
Substitute 2×A for B and F, 3×A for C, 6×A for D, and 4×A for E in eq.1: A + 2×A + 3×A + 6×A + 4×A + 2×A = 18 which simplifies to 18×A = 18 Divide both sides of the above equation by 18: 18×A ÷ 18 = 18 ÷ 18 which makes A = 1 making B = F = 2×A = 2 × 1 = 2 C = 3×A = 3 × 1 = 3 D = 6×A = 6 × 1 = 6 E = 4×A = 4 × 1 = 4 and ABCDEF = 123642