Puzzle for October 8, 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
Add F and C to both sides of eq.2: E – F + F + C = A – C + F + C which becomes E + C = A + F which may be written as C + E = A + F In eq.3, replace C + E with A + F: eq.3a) A + F = D
Hint #2
In eq.4, replace D with A + F (from eq.3a): A + F – E – F = A – B which becomes A – E = A – B Subtract A from each side of the equation above: A – E – A = A – B – A which makes –E = –B which means E = B
Hint #3
In eq.5, substitute B for E: B + C + F = D + B Subtract B from both sides of the above equation: B + C + F – B = D + B – B which becomes eq.5a) C + F = D
Hint #4
Substitute C + F for D (from eq.5a) in eq.3a: A + F = C + F Subtract F from each side of the above equation: A + F – F = C + F – F which makes A = C
Hint #5
Substitute A for C in eq.2: E – F = A – A which becomes E – F = 0 Add F to both sides of the equation above: E – F + F = 0 + F which makes E = F and also makes B = E = F
Hint #6
Substitute B for F in eq.3a: eq.3b) A + B = D
Hint #7
Substitute (A + B) for D (from eq.3b), B for E and F in eq.6: A + C – (A + B) – B = B + B + B which becomes A + C – A – B – B = 3×B which becomes C – 2×B = 3×B Add 2×B to both sides of the above equation: C – 2×B + 2×B = 3×B + 2×B which makes C = 5×B and also makes A = C = 5×B
Hint #8
Substitute 5×B for A in eq.3b: 5×B + B = D which becomes 6×B = D
Solution
Substitute 5×B for A and C, 6×B for D, and B for E and F in eq.1: 5×B + B + 5×B + 6×B + B + B = 19 which simplifies to 19×B = 19 Divide both sides of the equation above by 19: 19×B ÷ 19 = 19 ÷ 19 which means B = 1 making A = C = 5×B = 5 × 1 = 5 D = 6×B = 6 × 1 = 6 E = F = B = 1 and ABCDEF = 515611