Puzzle for October 30, 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.
* E! is E-factorial.
Scratchpad
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Hint #1
In eq.2, replace C with B + D (from eq.4): A + B = B + D + D which becomes A + B = B + 2×D Subtract B from each side of the equation above: A + B – B = B + 2×D – B which makes A = 2×D
Hint #2
In eq.3, replace A with 2×D: B + E = 2×D + D which becomes eq.3a) B + E = 3×D
Hint #3
In eq.5, replace A with 2×D: C – D = 2×D – F Add D and F to both sides of the equation above: C – D + D + F = 2×D – F + D + F which becomes eq.5a) C + F = 3×D
Hint #4
eq.1 may be re-written as: A + D + B + E + C + F = 27 In the above equation, substitute 2×D for A, 3×D for B + E (from eq.3a), and 3×D for C + F (from eq.5a): 2×D + D + 3×D + 3×D = 27 which makes 9×D = 27 Divide both sides by 9: 9×D ÷ 9 = 27 ÷ 9 which makes D = 3 making A = 2×D = 2 × 3 = 6
Hint #5
Substitute 6 for A, and 3 for D in eq.6: E! = 6 × 3 × (6 + ((6 ÷ 3) ÷ 3)) which becomes E! = 18 × 6⅔ which becomes E! = 120 which is equivalent to E! = 1 x 2 x 3 x 4 x 5 which means E = 5
Hint #6
Substitute 5 for E, and 3 for D in eq.3a: B + 5 = 3×3 which becomes B + 5 = 9 Subtract 5 from each side of the equation above: B + 5 – 5 = 9 – 5 which makes B = 4
Hint #7
Substitute 4 for B, and 3 for D in eq.4: C = 4 + 3 which makes C = 7
Solution
Substitute 7 for C, and 3 for D in eq.5a: 7 + F = 3×3 which becomes 7 + F = 9 Subtract 7 from each side of the above equation: 7 + F – 7 = 9 – 7 which makes F = 2 making ABCDEF = 647352