Puzzle for October 4, 2019 ( )
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, add F to both sides, and subtract A from both sides: A + F – A = B – F + F – A which becomes eq.5a) F = B – A In eq.2, replace F with B – A (from eq.5a): E + B – A = A + B In the above equation, add A to both sides, and subtract B from both sides: E + B – A + A – B = A + B + A – B which simplifies to E = 2×A
Hint #2
In eq.3, replace A + B with E + F (from eq.2): C + D = E + F – F which becomes eq.3a) C + D = E
Hint #3
In eq.4, substitute C + D for E (from eq.3a): D – A = C + D – B + F In the equation above, subtract D from each side, and add B to each side: D – A – D + B = C + D – B + F – D + B which becomes –A + B = C + F which may be written as eq.4a) B – A = C + F
Hint #4
In eq.4a, substitute F for B – A (from eq.5a): F = C + F Subtract F from both sides of the above equation: F – F = C + F – F which means 0 = C
Hint #5
Substitute B – A for F (from eq.5a) in eq.4: D – A = E – B + B – A which becomes D – A = E – A Add A to both sides of the equation above: D – A + A = E – A + A which makes D = E and also makes D = E = 2×A
Hint #6
Add B, C, and F to both sides of eq.6: B – C + F + B + C + F = A – B + D – F + B + C + F which becomes 2×B + 2×F = A + C + D Substitute 0 for C, and 2×A for D in the equation above: 2×B + 2×F = A + 0 + 2×A which becomes eq.6a) 2×(B + F) = 3×A
Hint #7
Substitute B – A for F (from eq.5a) in eq.6a: 2×(B + B – A) = 3×A which becomes 2×(2×B – A) = 3×A which becomes 4×B – 2×A = 3×A Add 2×A to both sides of the above equation: 4×B – 2×A + 2×A = 3×A + 2×A which makes 4×B = 5×A Divide both sides by 4: 4×B ÷ 4 = 5×A ÷ 4 which makes B = 1¼×A
Hint #8
Substitute 1¼×A for B in eq.5a: F = 1¼×A – A which makes F = ¼×A
Solution
Substitute 1¼×A for B, 0 for C, and 2×A for D and E, and ¼×A for F in eq.1: A + 1¼×A + 0 + 2×A + 2×A + ¼×A = 26 which simplifies to 6½×A = 26 Divide both sides of the equation above by 6½: 6½×A ÷ 6½ = 26 ÷ 6½ which means A = 4 making B = 1¼×A = 1¼ × 4 = 5 D = E = 2×A = 2 × 4 = 8 F = ¼×A = ¼ × 4 = 1 and ABCDEF = 450881