Puzzle for January 11, 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
In eq.6, replace A + D with C (from eq.3): B – C – E = C + E – F Add C, E, and F to both sides of the above equation: B – C – E + C + E + F = C + E – F + C + E + F which becomes B + F = 2×C + 2×E which may be written as eq.6a) B + F = 2×(C + E)
Hint #2
In eq.6a, replace C + E with B (from eq.2): B + F = 2×(B) which is simply the same as B + F = 2×B Subtract B from each side of the equation above: B + F – B = 2×B – B which makes F = B
Hint #3
In eq.4, substitute B for F: B + D = A + B Subtract B from each side of the above equation: B + D – B = A + B – B which makes D = A
Hint #4
Substitute A for D into eq.3: A + A = C which makes 2×A = C
Hint #5
Substitute 2×A for C, and A for D in eq.5: E – A – 2×A = A + 2×A + A which becomes E – 3×A = 4×A Add 3×A to both sides of the above equation: E – 3×A + 3×A = 4×A + 3×A which makes E = 7×A
Hint #6
Substitute 2×A for C, and 7×A for E in eq.2: 2×A + 7×A = B which makes 9×A = B and also makes F = B = 9×A
Solution
Substitute 9×A for B and F, 2×A for C, A for D, and 7×A for E in eq.1: A + 9×A + 2×A + A + 7×A + 9×A = 29 which simplifies to 29×A = 29 Divide both sides of the above equation by 29: 29×A ÷ 29 = 29 ÷ 29 which means A = 1 making B = F = 9×A = 9 × 1 = 9 C = 2×A = 2 × 1 = 2 D = A = 1 E = 7×A = 7 × 1 = 7 and ABCDEF = 192179