Puzzle for October 22, 2021 ( )
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
Subtract the left and right sides of eq.4 from the left and right sides of eq.5, respectively: C + D – F – (D – B) = B + F – (B – C + E) which becomes C + D – F – D + B = B + F – B + C – E which becomes C – F + B = F + C – E In the above equation, add F and E to both sides, and subtract C from both sides: C – F + B + F + E – C = F + C – E + F + E – C which simplifies to eq.5a) B + E = 2×F
Hint #2
In eq.5a, replace B + E with A (from eq.2): eq.5b) A = 2×F
Hint #3
In eq.6, replace A with 2×F (from eq.5b): D + E – F = 2×F + F which becomes D + E – F = 3×F Add F to both sides of the above equation: D + E – F + F = 3×F + F which becomes D + E = 4×F which may be written as eq.6a) D + E = 2×(2×F)
Hint #4
In eq.6a, substitute B + E for 2×F (from eq.5a): D + E = 2×(B + E) which becomes D + E = 2×B + 2×E Subtract E from both sides of the above equation: D + E – E = 2×B + 2×E – E which becomes eq.6b) D = 2×B + E
Hint #5
Substitute 2×B + E for D (from eq.6b), and B + E for A (from eq.2) in eq.3: B + 2×B + E = B + E + E which becomes 3×B + E = B + 2×E Subtract E and B from each side of the above equation: 3×B + E – E – B = B + 2×E – E – B which makes 2×B = E
Hint #6
Substitute 2×B for E in eq.6b: D = 2×B + 2×B which makes D = 4×B
Hint #7
Substitute 2×B for E in eq.5a: B + 2×B = 2×F which makes 3×B = 2×F Divide both sides of the above equation by 2: 3×B ÷ 2 = 2×F ÷ 2 which makes 1½×B = F
Hint #8
Substitute (1½×B) for F into eq.5b: A = 2×(1½×B) which makes A = 3×B
Hint #9
Substitute 4×B for D, and 1½×B for F in eq.5: C + 4×B – 1½×B = B + 1½×B which becomes C + 2½×B = 2½×B Subtract 2½×B from both sides of the equation above: C + 2½×B – 2½×B = 2½×B – 2½×B which makes C = 0
Solution
Substitute 3×B for A, 0 for C, 4×B for D, 2×B for E, and 1½×B for F in eq.1: 3×B + B + 0 + 4×B + 2×B + 1½×B = 23 which simplifies to 11½×B = 23 Divide both sides of the above equation by 11½: 11½×B ÷ 11½ = 23 ÷ 11½ which means B = 2 making A = 3×B = 3 × 2 = 6 D = 4×B = 4 × 2 = 8 E = 2×B = 2 × 2 = 4 F = 1½×B = 1½ × 2 = 3 and ABCDEF = 620843