Puzzle for February 26, 2020 ( )
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.5, replace A with C + F (from eq.6): E – D = C + F – C which becomes E – D = F In the above equation, add D to both sides, and subtract F from each side: E – D + D – F = F + D – F which makes eq.5a) E – F = D
Hint #2
In eq.3, replace D with E – F (rom eq.5a): C + E – F – F = E + F which becomes C + E – 2×F = E + F In the above equation, add 2×F to both sides, and subtract E from each side: C + E – 2×F + 2×F – E = E + F + 2×F – E which simplifies to C = 3×F
Hint #3
In eq.6, substitute 3×F for C: 3×F + F = A which makes 4×F = A
Hint #4
Substitute 4×F for A, and 3×F for C in eq.4: 4×F + B = 3×F + D + F which becomes 4×F + B = 4×F + D Subtract 4×F from both sides of the above equation: 4×F + B – 4×F = 4×F + D – 4×F which makes B = D
Hint #5
Substitute B for D, and 4×F for A in eq.2: B + B = 4×F + F which means 2×B = 5×F Divide both sides by 2: 2×B ÷ 2 = 5×F ÷ 2 which makes B = 2½×F which also makes D = B = 2½×F
Hint #6
Substitute 2½×F for D in eq.5a: E – F = 2½×F Add F to both sides: E – F + F = 2½×F + F which makes E = 3½×F
Solution
Substitute 4×F for A, 2½×F for B and D, 3×F for C, and 3½×F for E in eq.1: 4×F + 2½×F + 3×F + 2½×F + 3½×F + F = 33 which simplifies to 16½×F = 33 Divide both sides of the equation above by 16½: 16½×F ÷ 16½ = 33 ÷ 16½ which means F = 2 making A = 4×F = 4 × 2 = 8 B = D = 2½×F = 2½ × 2 = 5 C = 3×F = 3 × 2 = 6 E = 3½×F = 3½ × 2 = 7 and ABCDEF = 856572