Puzzle for November 20, 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 E + F with A + C (from eq.2): D + A + C = A + B + C Subtract A and C from both sides of the above equation: D + A + C – A – C = A + B + C – A – C which simplifies to D = B
Hint #2
Add D to each side of eq.6: A + D = C – D + D which becomes A + D = C In the equation above, replace D with B: eq.6a) A + B = C
Hint #3
In eq.4, substitute A + B for C (from eq.6a), and substitute B for D: A + B + E = B + B Subtract A and B from both sides of the equation above: A + B + E – A – B = B + B – A – B which becomes eq.4a) E = B – A
Hint #4
In eq.3, substitute B for D, B – A for E (from eq.4a), and A + B for C (from eq.6a): B + B – A + F = A + B + A + B which becomes 2×B – A + F = 2×A + 2×B In the above equation, add A to each side, and subtract 2×B from each side: 2×B – A + F + A – 2×B = 2×A + 2×B + A – 2×B which makes F = 3×A
Hint #5
Substitute 3×A for F in eq.5: B – A = A + 3×A which becomes B – A = 4×A Add A to both sides of the equation above: B – A + A = 4×A + A which makes B = 5×A and also makes D = B = 5×A
Hint #6
Substitute 5×A for B in eq.6a: A + 5×A = C which makes 6×A = C
Hint #7
Substitute 5×A for B in eq.4a: E = 5×A – A which makes E = 4×A
Solution
Substitute 5×A for B and D, 6×A for C, 4×A for E, and 3×A for F in eq.1: A + 5×A + 6×A + 5×A + 4×A + 3×A = 24 which simplifies to 24×A = 24 Divide both sides of the equation above by 24: 24×A ÷ 24 = 24 ÷ 24 which means A = 1 making B = D = 5×A = 5 × 1 = 5 C = 6×A = 6 × 1 = 6 E = 4×A = 4 × 1 = 4 F = 3×A = 3 × 1 = 3 and ABCDEF = 156543