Puzzle for February 24, 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
In eq.2, replace C with B – A (from eq.4): E = A + B – A which makes E = B
Hint #2
In eq.6, replace C with B – A (from eq.4): A + B = B – A + D In the above equation, subtract B from both sides, and add A to each side: A + B – B + A = B – A + D – B + A which makes 2×A = D
Hint #3
eq.5 may be written as: B + E = C + F + D In the above equation, replace C + F with D + E (from eq.3): B + E = D + E + D which becomes B + E = 2×D + E Subtract E from each side: B + E – E = 2×D + E – E which makes B = 2×D Replace D with (2×A): B = 2×(2×A) which makes B = 4×A and also makes E = B = 4×A
Hint #4
In eq.4, substitute 4×A for B: C = 4×A – A which makes C = 3×A
Hint #5
Substitute 2×A for D, 4×A for E, and 3×A for C in eq.3: 2×A + 4×A = 3×A + F which becomes 6×A = 3×A + F Subtract 3×A from both sides of the above equation: 6×A – 3×A = 3×A + F – 3×A which makes 3×A = F
Solution
Substitute 4×A for B and E, 3×A for C and F, and 2×A for D in eq.1: A + 4×A + 3×A + 2×A + 4×A + 3×A = 17 which simplifies to 17×A = 17 Divide both sides of the equation above by 17: 17×A ÷ 17 = 17 ÷ 17 which means A = 1 making B = E = 4×A = 4 × 1 = 4 C = F = 3×A = 3 × 1 = 3 D = 2×A = 2 × 1 = 2 and A = 143243