Puzzle for April 1, 2019 ( )
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.2, replace A with E + F (from eq.5): E + F + B = C + E Subtract E from each side of the equation above: E + F + B - E = C + E - E which becomes F + B = C In eq.3, replace C with F + B: B + E = F + B + F which becomes B + E = 2×F + B Subtract B from both sides of the equation above: B + E - B = 2×F + B - B which makes E = 2×F
Hint #2
In eq.5, substitute 2×F for E: 2×F + F = A which makes 3×F = A
Hint #3
In eq.6, substitute 3×F for A: D - 3×F = 3×F + F which becomes D - 3×F = 4×F Add 3×F to each side of the above equation: D - 3×F + 3×F = 4×F + 3×F which means D = 7×F
Hint #4
Substitute 3×F for A, 7×F for D, and 2×F for E in eq.4: 3×F + C = 7×F - 2×F - F which becomes 3×F + C = 4×F Subtract 3×F from both sides: 3×F + C - 3×F = 4×F - 3×F which makes C = F
Hint #5
Substitute 3×F for A, F for C, and 2×F for E in eq.2: 3×F + B = F + 2×F which becomes 3×F + B = 3×F Subtract 3×F from both sides: 3×F + B - 3×F = 3×F - 3×F which means B = 0
Solution
Substitute 3×F for A, 0 for B, F for C, 7×F for D, and 2×F for E in eq.1: 3×F + 0 + F + 7×F + 2×F + F = 14 which becomes 14×F = 14 Divide both sides by 14: 14×F ÷ 14 = 14 ÷ 14 which means F = 1 making A = 3×F = 3 × 1 = 3 C = F = 1 D = 7×F = 7 × 1 = 7 E = 2×F = 2 × 1 = 2 and ABCDEF = 301721