Puzzle for October 22, 2022 ( )
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
eq.5 may be expressed as: B = (A + C + D + E + F) ÷ 5 Multiply both sides of the above equation by 5: 5 × B = 5 × (A + C + D + E + F) ÷ 5 which becomes eq.5a) 5×B = A + C + D + E + F
Hint #2
eq.1 may be written as: A + C + D + E + F + B = 36 In the equation above, replace A + C + D + E + F with 5×B (from eq.5a): 5×B + B = 36 which makes 6×B = 36 Divide both sides by 6: 6×B ÷ 6 = 36 ÷ 6 which makes B = 6
Hint #3
eq.4 may be written as: D + F - B = A + E + B - D In the above equation, replace A + E with B + D - A (from eq.2): D + F - B = B + D - A + B - D which becomes eq.4a) D + F - B = 2×B - A
Hint #4
Add B and A to both sides of eq.4a: D + F - B + B + A = 2×B - A + B + A which becomes D + F + A = 3×B In the equation above, substitute 6 for B: D + F + A = 3×6 which becomes D + F + A = 18 which may be written as eq.4b) A + D + F = 18
Hint #5
In eq.3, substitute A + E for B + D - A (from eq.2): C - D + E = A + E In the above equation, add D to both sides, and subtract E from both sides: C - D + E + D - E = A + E + D - E which becomes eq.3a) C = A + D
Hint #6
In eq.4b, substitute C for A + D (from eq.3a): C + F = 18 Since C and F must be one-digit non-negative integers, the above equation makes: C = 9 and F = 9
Hint #7
eq.5a may be re-written as: 5×B = A + D + F + C + E Substitute 6 for B, 18 for A + D + F (from eq.4b), and 9 for C in the above equation: 5×6 = 18 + 9 + E which becomes 30 = 27 + E Subtract 27 from each side: 30 - 27 = 27 + E - 27 which makes 3 = E
Hint #8
Substitute 9 for C in eq.3a: 9 = A + D Subtract A from both sides of the above equation: 9 - A = A + D - A which becomes eq.3b) 9 - A = D
Hint #9
Substitute 3 for E, 6 for B, and 9 - A for D (from eq.3b) in eq.2: A + 3 = 6 + 9 - A - A which becomes A + 3 = 15 - 2×A In the above equation, subtract 3 from both sides, and add 2×A to both sides: 3 + A - 3 + 2×A = 15 - 2×A - 3 + 2×A which makes 3×A = 12 Divide both sides by 3: 3×A ÷ 3 = 12 ÷ 3 which makes A = 4
Solution
Substitute 4 for A in eq.3b: 9 - 4 = D which makes 5 = D and makes ABCDEF = 469539