Puzzle for October 9, 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
Add the left and right sides of eq.6 to the left and right sides of eq.5, respectively: C – A – B – F + A + F = D + F + D – B – F which becomes C – B = 2×D – B Add B to both sides of the above equation: C – B + B = 2×D – B + B which makes eq.5a) C = 2×D
Hint #2
In eq.2, replace C with 2×D: 2×D = D + E Subtract D from both sides of the equation above: 2×D – D = D + E – D which makes D = E
Hint #3
In eq.4, substitute D for E: D – D = B – F which becomes 0 = B – F Add F to both sides of the equation above: 0 + F = B – F + F which makes F = B
Hint #4
Substitute D for E in eq.3: B + D = A + D Subtract D from both sides of the above equation: B + D – D = A + D – D which makes B = A and makes F = B = A
Hint #5
Substitute A for B and F in eq.6: A + A = D – A – A which becomes 2×A = D – 2×A Add 2×A to both sides of the above equation: 2×A + 2×A = D – 2×A + 2×A which makes 4×A = D and also makes E = D = 4×A
Hint #6
Substitute (4×A) for D in eq.5a: C = 2×(4×A) which makes C = 8×A
Solution
Substitute A for B and F, 8×A for C, and 4×A for D and E in eq.1: A + A + 8×A + 4×A + 4×A + A = 19 which simplifies to 19×A = 19 Divide both sides of the above equation by 19: 19×A ÷ 19 = 19 ÷ 19 which means A = 1 making B = F = A = 1 C = 8×A = 8 × 1 = 8 D = E = 4×A = 4 × 1 = 4 and ABCDEF = 118441