Puzzle for January 10, 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
In eq.3, replace B with D + F (from eq.2): A = D + F + F which becomes eq.3a) A = D + 2×F
Hint #2
In eq.6, replace A with D + 2×F (from eq.3a): D + E = D + 2×F + F which becomes D + E = D + 3×F Subtract D from each side of the equation above: D + E - D = D + 3×F - D which makes E = 3×F
Hint #3
In eq.4, substitute 3×F for E: C = 3×F + F which makes C = 4×F
Hint #4
Substitute 4×F for C, and 3×F for E in eq.5: B = 4×F + 3×F which makes B = 7×F
Hint #5
Substitute 7×F for B in eq.2: 7×F = D + F Subtract F from each side of the above equation: 7×F - F = D + F - F which makes 6×F = D
Hint #6
Substitute 6×F for D in eq.3a: A = 6×F + 2×F which becomes A = 8×F
Solution
Substitute 8×F for A, 7×F for B, 4×F for C, 6×F for D, and 3×F for E in eq.1: 8×F + 7×F + 4×F + F + 6×F + 3×F = 29 which simplifies to 29×F = 29 Divide both sides of the above equation by 29: 29×F ÷ 29 = 29 ÷ 29 which means F = 1 making A = 8×F = 8 × 1 = 8 B = 7×F = 7 × 1 = 7 C = 4×F = 4 × 1 = 4 D = 6×F = 6 × 1 = 6 E = 3×F = 3 × 1 = 3 and ABCDEF = 874631