Puzzle for October 18, 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 positive integer.
Scratchpad
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Hint #1
Add F to both sides of eq.2: D - F + F = B + E + F which becomes D = B + E + F which may be written as D = B + F + E In the above equation, replace B + F with E (from eq.1): D = E + E which makes eq.2a) D = 2×E
Hint #2
In eq.6, replace D with 2×E: 2×E = E × F Divide both sides of the above equation by E: 2×E ÷ E = E × F ÷ E which makes 2 = F
Hint #3
In eq.5, substitute 2 for F: E = B × 2 which may be written as eq.5a) E = 2×B
Hint #4
Substitute 2×B for E, and 2 for F in eq.1: 2×B = B + 2 Subtract B from each side of the equation above: 2×B - B = B + 2 - B which makes B = 2
Hint #5
Substitute 2 for B in eq.5a: E = 2×2 which makes E = 4
Hint #6
Substitute 4 for E in eq.2a: D = 2×4 which makes D = 8
Hint #7
Substitute 2 for F and B in eq.3: A + 2 = 2 + C Subtract 2 from each side of the above equation: A + 2 - 2 = 2 + C - 2 which makes A = C
Solution
Substitute 8 for D, 2 for F, and A for C in eq.4: 8 + 2 = A + A - 8 which becomes 10 = 2×A - 8 Add 8 to both sides of the equation above: 10 + 8 = 2×A - 8 + 8 which makes 18 = 2×A Divide both sides by 2: 18 ÷ 2 = 2×A ÷ 2 which makes 9 = A and also makes A = C = 9 and makes ABCDEF = 929842