Puzzle for January 16, 2023 ( )
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.5, replace A + C with D + E (from eq.4): D + E + F = B + D + E Subtract D and E from each side of the equation above: D + E + F - D - E = B + D + E - D - E which simplifies to F = B
Hint #2
In eq.1, replace F with B: A = B + B which makes A = 2×B
Hint #3
In eq.2, replace F with B: C - B = B - B which becomes C - B = 0 Add B to both sides of the above equation: C - B + B = 0 + B which makes C = B
Hint #4
In eq.6, substitute B for C: D = B ÷ B which makes D = 1
Hint #5
Substitute 1 for D in eq.3: E - 1 = A + 1 Add 1 to both sides of the above equation: E - 1 + 1 = A + 1 + 1 which makes eq.3a) E = A + 2
Hint #6
Substitute B for C, 1 for D, and A + 2 for E (from eq.3a) in eq.4: A + B = 1 + A + 2 which becomes A + B = 3 + A Subtract A from each side of the above equation: A + B - A = 3 + A - A which makes B = 3 and also makes C = B = F = 3
Solution
Since B = 3, then: A = 2×B = 2×3 = 6 E = A + 2 = 6 + 2 = 8 (from eq.3a) and ABCDEF = 633183