Puzzle for September 19, 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 positive integer.
Scratchpad
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Hint #1
Add A and E to both sides of eq.5: C – A + A + E = D – E + A + E which becomes C + E = D + A which may be written as C + E = A + D In eq.3, replace A + D with C + E: eq.3a) F = C + E
Hint #2
In eq.3a, replace E with B – C (from eq.2): F = C + B – C which makes F = B
Hint #3
In eq.4, substitute B for F: E + B = A + B Subtract B from each side of the equation above: E + B – B = A + B – B which makes E = A
Hint #4
In eq.5, substitute A for E: C – A = D – A Add A to both sides of the above equation: C – A + A = D – A + A which makes C = D
Hint #5
Add C to both sides of eq.2: B – C + C = E + C which becomes eq.2a) B = E + C Add the left and right sides of eq.2a to the left and right sides of eq.3, respectively: F + B = A + D + E + C Substitute B for F in the equation above: B + B = A + D + E + C which may be written as eq.3b) 2×B = A + C + D + E
Hint #6
eq.1 may be written as: A + C + D + E + B + F = 28 Substitute B for F, and 2×B for A + C + D + E (from eq.3b) in the above equation: 2×B + B + B = 28 which becomes 4×B = 28 Divide both sides by 4: 4×B ÷ 4 = 28 ÷ 4 which makes B = 7 and also makes F = B = 7
Hint #7
Substitute A for E, 7 for B and F, and C for D in eq.6: A × A = 7 – C – C + 7 which becomes eq.6a) A² = 14 – 2×C
Hint #8
Substitute 7 for B, and A for E in eq.2: 7 – C = A In the equation above, add C to both sides, and subtract A from both sides: 7 – C + C – A = A + C – A which becomes eq.2b) 7 – A = C
Hint #9
Substitute (7 – A) for C (from eq.2b) in eq.6a: A² = 14 – 2×(7 – A) which is equivalent to A² = 14 – 14 + 2×A which becomes A² = 2×A Divide both sides of the equation above by A: A² ÷ A = 2×A ÷ A which makes A = 2 and also makes E = A = 2
Solution
Substitute 2 for A in eq.2b: 7 – 2 = C which makes 5 = C and also makes D = C = 5 making ABCDEF = 275527