Puzzle for October 8, 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 F with C + E (from eq.4): D + C + E = B which may be written as C + D + E = B In eq.3, replace C + D + E with B: A = B + B which makes eq.3a) A = 2×B
Hint #2
In eq.4, substitute B – E for C + E (from eq.6): F = B – E Add E to both sides of the above equation: F + E = B – E + E which becomes eq.4a) F + E = B
Hint #3
Substitute F + E for B (from eq.4a) in eq.2: D + F = F + E Subtract F from each side of the above equation: D + F – F = F + E – F which makes D = E
Hint #4
Substitute E for D, and C + E for F (from eq.4) in eq.5: B + C – E = E + E + C + E which becomes B + C – E = C + 3×E In the above equation, subtract C from each side, and add E to each side: B + C – E – C + E = C + 3×E – C + E which simplifies to B = 4×E
Hint #5
Substitute (4×E) for B in eq.3a: A = 2×(4×E) which makes A = 8×E
Hint #6
Substitute 4×E for B in eq.4a: F + E = 4×E Subtract E from each side of the above equation: F + E – E = 4×E – E which makes F = 3×E
Hint #7
Substitute 3×E for F in eq.4: 3×E = C + E Subtract E from each side of the above equation: 3×E – E = C + E – E which makes 2×E = C
Solution
Substitute 8×E for A, 4×E for B, 2×E for C, E for D, and 3×E for F in eq.1: 8×E + 4×E + 2×E + E + E + 3×E = 19 which simplifies to 19×E = 19 Divide both sides of the equation above by 19: 19×E ÷ 19 = 19 ÷ 19 which means E = 1 making A = 8×E = 8 × 1 = 8 B = 4×E = 4 × 1 = 4 C = 2×E = 2 × 1 = 2 D = E = 1 F = 3×E = 3 × 1 = 3 and ABCDEF = 842113