Puzzle for August 10, 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.3, replace A with B + E (from eq.4): B + F = B + E + D Subtract B from both sides of the above equation: B + F – B = B + E + D – B which becomes eq.3a) F = E + D
Hint #2
In eq.6, replace F with E + D (from eq.3a): C + D = A + E + D Subtract D from each side of the equation above: C + D – D = A + E + D – D which becomes eq.6a) C = A + E
Hint #3
In eq.2, substitute A + E for C (from eq.6a): E + F = B + A + E Subtract E from each side of the above equation: E + F – E = B + A + E – E which becomes eq.2a) F = B + A
Hint #4
Substitute B + A for F (from eq.2a) in eq.3: B + B + A = A + D which becomes 2×B + A = A + D Subtract A from each side of the equation above: 2×B + A – A = A + D – A which makes eq.3b) 2×B = D
Hint #5
Substitute 2×B for D (from eq.3b) in eq.5: B + 2×B = C + E which means eq.5a) 3×B = C + E
Hint #6
Substitute B + E for A (from eq.4) in eq.6a: C = B + E + E which makes eq.6b) C = B + 2×E
Hint #7
Substitute B + 2×E for C (from eq.6b) in eq.5a: 3×B = B + 2×E + E which becomes 3×B = B + 3×E Subtract B from each side of the above equation: 3×B – B = B + 3×E – B which means 2×B = 3×E Divide both sides by 2: 2×B ÷ 2 = 3×E ÷ 2 which makes B = 1½×E
Hint #8
Substitute 1½×E for B in eq.4: A = 1½×E + E which means A = 2½×E
Hint #9
Substitute 1½×E for B in eq.6b: C = 1½×E + 2×E which makes C = 3½×E
Hint #10
Substitute (1½×E) for B in eq.3b: 2×(1½×E) = D which means 3×E = D
Hint #11
Substitute 3×E for D in eq.3a: F = E + 3×E which makes F = 4×E
Solution
Substitute 2½×E for A, 1½×E for B, 3½×E for C, 3×E for D, and 4×E for F in eq.1: 2½×E + 1½×E + 3½×E + 3×E + E + 4×E = 31 which simplifies to 15½×E = 31 Divide both sides of the equation above by 15½: 15½×E ÷ 15½ = 31 ÷ 15½ which means E = 2 making A = 2½×E = 2½ × 2 = 5 B = 1½×E = 1½ × 2 = 3 C = 3½×E = 3½ × 2 = 7 D = 3×E = 3 × 2 = 6 F = 4×E = 4 × 2 = 8 and ABCDEF = 537628