Puzzle for August 26, 2021 ( )
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
Subtract the left and right sides of eq.5 from the left and right sides of eq.6, respectively: A + C – E – (C – D) = D + E + F – (D + E – F) which becomes A + C – E – C + D = D + E + F – D – E + F which becomes A – E + D = 2×F Add E to both sides of the above equation: A – E + D + E = 2×F + E which becomes eq.6a) A + D = 2×F + E
Hint #2
In eq.6a, replace A + D with B + E (from eq.3): B + E = 2×F + E Subtract E from each side of the above equation: B + E – E = 2×F + E – E which becomes B = 2×F
Hint #3
In eq.2, substitute 2×F for B: C = 2×F + F which makes C = 3×F
Hint #4
In eq.4, substitute 2×F for B: D + E = A + 2×F – D Add D to both sides of the above equation: D + E + D = A + 2×F – D + D which becomes eq.4a) 2×D + E = A + 2×F
Hint #5
Substitute 3×F for C in eq.5: 3×F – D = D + E – F Add D and F to both sides of the above equation: 3×F – D + D + F = D + E – F + D + F which becomes eq.5a) 4×F = 2×D + E
Hint #6
In eq.4a, replace 2×D + E with 4×F (from eq.5a): 4×F = A + 2×F Subtract 2×F from each side of the above equation: 4×F – 2×F = A + 2×F – 2×F which makes 2×F = A
Hint #7
In eq.6a, substitute A for 2×F: A + D = A + E Subtract A from both sides of the above equation: A + D – A = A + E – A which makes D = E
Hint #8
Substitute D for E in eq.5a: 4×F = 2×D + D which becomes 4×F = 3×D Divide both sides of the above equation by 3: 4×F ÷ 3 = 3×E ÷ 3 which makes 1⅓×F = E and also makes D = E = 1⅓×F
Solution
Substitute 2×F for A and B, 3×F for C, and 1⅓×F for D and E in eq.1: 2×F + 2×F + 3×F + 1⅓×F + 1⅓×F + F = 32 which simplifies to 10⅔×F = 32 Divide both sides of the above equation by 10⅔: 10⅔×F ÷ 10⅔ = 32 ÷ 10⅔ which means F = 3 making A = B = 2×F = 2 × 3 = 6 C = 3×F = 3 × 3 = 9 D = E = 1⅓×F = 1⅓ × 3 = 4 and ABCDEF = 669443