Puzzle for August 2, 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.4, replace B with A + F (from eq.2): A + D = A + F + F which becomes A + D = A + 2×F Subtract A from each side of the equation above: A + D – A = A + 2×F – A which makes D = 2×F
Hint #2
In eq.3, replace D with 2×F: C = 2×F + F which makes C = 3×F
Hint #3
In eq.5, substitute 3×F for C, and 2×F for D: E + F = 3×F + 2×F – F which makes E + F = 4×F Subtract F from each side of the above equation: E + F – F = 4×F – F which makes E = 3×F
Hint #4
Substitute 3×F for C and E, and A + F for B (from eq.2) in eq.6: 3×F – A = A + A + F – 3×F which becomes 3×F – A = 2×A – 2×F Add A and 2×F to both sides of the above equation: 3×F – A + A + 2×F = 2×A – 2×F + A + 2×F which becomes 5×F = 3×A Divide both sides by 3: 5×F ÷ 3 = 3×A ÷ 3 which makes 1⅔×F = A
Hint #5
Substitute 1⅔×F for A in eq.2: B = 1⅔×F + F which makes B = 2⅔×F
Solution
Substitute 1⅔×F for A, 2⅔×F for B, 3×F for C and E, and 2×F for D in eq.1: 1⅔×F + 2⅔×F + 3×F + 2×F + 3×F + F = 40 which simplifies to 13⅓×F = 40 Divide both sides of the above equation by 13⅓: 13⅓×F ÷ 13⅓ = 40 ÷ 13⅓ which means F = 3 making A = 1⅔×F = 1⅔ × 3 = 5 B = 2⅔×F = 2⅔ × 3 = 8 C = E = 3×F = 3 × 3 = 9 D = 2×F = 2 × 3 = 6 and ABCDEF = 589693