Puzzle for January 9, 2021 ( )
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 A – B with C – A (from eq.5): B + C = C – A + E In the above equation, subtract C from both sides, and add A to both sides: B + C – C + A = C – A + E – C + A which becomes B + A = E which may be written as eq.3a) A + B = E
Hint #2
In eq.2, replace A + B with E (from eq.3a): E + F = E Subtract E from each side of the equation above: E + F – E = E – E which means F = 0
Hint #3
Subtract the left and right sides of eq.4 from the left and right sides of eq.3, respectively: B + C – (A + C) = A – B + E – (D + E – B) which is equivalent to B + C – A – C = A – B + E – D – E + B which becomes B – A = A – D Add A and D to both sides of the equation above: B – A + A + D = A – D + A + D which becomes eq.3b) B + D = 2×A
Hint #4
Add A and B to both sides of eq.5: C – A + A + B = A – B + A + B which becomes C + B = 2×A which may be written as eq.5a) B + C = 2×A In eq.3b, substitute B + C for 2×A (from eq.5a): B + D = B + C Subtract B from both sides of the equation above: B + D – B = B + C – B which makes D = C
Hint #5
Add D to both sides of eq.2: E + F + D = A + B + D In the above equation, substitute 2×A for B + D (from eq.3b): E + F + D = A + 2×A which becomes E + F + D = 3×A which may be written as eq.2a) D + E + F = 3×A
Hint #6
Substitute 2×A for B + C (from eq.5a), and 3×A for D + E + F (from eq.2a) in eq.1: A + 2×A + 3×A = 30 which makes 6×A = 30 Divide both sides of the equation above by 6: 6×A ÷ 6 = 30 ÷ 6 which makes A = 5
Hint #7
Substitute 5 for A in eq.5a: B + C = 2×5 which becomes B + C = 10 Subtract B from each side of the equation above: B + C – B = 10 – B which becomes C = 10 – B and also makes eq.5b) D = C = 10 – B
Hint #8
Substitute 5 for A in eq.3a: eq.3c) 5 + B = E
Solution
Substitute 10 – B for D (from eq.5b), 5 + B for E (from eq.3c), 0 for F, and 5 for A in eq.6: 10 – B + 5 + B + 0 = 5 × B which makes 15 = 5×B Divide both sides of the equation above by 5: 15 ÷ 5 = 5×B ÷ 5 which means 3 = B making D = C = 10 – B = 10 – 3 = 7 (from eq.5b) E = 5 + B = 5 + 3 = 8 (from eq.3c) and ABCDEF = 537780