Puzzle for December 31, 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.6, replace A + C with D + F (from eq.3), and B + E with D + F (from eq.2): D + F + D = D + F + F Subtract D and F from each side of the above equation: D + F + D – D – F = D + F + F – D – F which makes D = F
Hint #2
In eq.3, replace F with D: D + D = A + C which becomes eq.3a) 2×D = A + C
Hint #3
In eq.2, replace F with D: B + E = D + D which becomes eq.2a) B + E = 2×D
Hint #4
eq.1 may be written as: A + C + D + B + E + F = 36 In the equation above, substitute 2×D for A + C (from eq.3a) and for B + E (from eq.2a), and D for F: 2×D + D + 2×D + D = 36 which becomes 6×D = 36 Divide both sides of the above equation by 6: 6×D ÷ 6 = 36 ÷ 6 which makes D = 6 and also makes F = D = 6
Hint #5
In eq.2, substitute A + C for D + F (from eq.3): B + E = A + C Subtract C from both sides of the equation above: B + E – C = A + C – C which becomes eq.2b) B + E – C = A
Hint #6
Substitute B + E – C for A (from eq.2b), and 6 for F in eq.4: B + E – C + 6 = B + C + E In the equation above, subtract B and E from both sides, and add C to both sides: B + E – C + 6 – B – E + C = B + C + E – B – E + C which simplifies to 6 = 2×C Divide both sides by 2: 6 ÷ 2 = 2×C ÷ 2 which makes 3 = C
Hint #7
Substitute 6 for D and F, and 3 for C in eq.3: 6 + 6 = A + 3 which becomes 12 = A + 3 Subtract 3 from each side of the above equation: 12 – 3 = A + 3 – 3 which makes 9 = A
Hint #8
Substitute 6 for F and D, and 9 for A in eq.5: B + 6 = 9 – B + 6 + E which becomes B + 6 = 15 – B + E In the above equation, subtract 15 from both sides, and add B to both sides: B + 6 – 15 + B = 15 – B + E – 15 + B which becomes eq.5a) 2×B – 9 = E
Hint #9
Substitute 2×B – 9 for E (from eq.5a), and 6 for D in eq.2a: B + 2×B – 9 = 2×6 which becomes 3×B – 9 = 12 Add 9 to both sides of the equation above: 3×B – 9 + 9 = 12 + 9 which becomes 3×B = 21 Divide both sides by 3: 3×B ÷ 3 = 21 ÷ 3 which makes B = 7
Solution
Substitute 7 for B in eq.5a: 2×7 – 9 = E which becomes 14 – 9 = E which makes 5 = E and makes ABCDEF = 973656