Puzzle for September 18, 2020 ( )
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.3, add F to both sides, and subtract A from both sides: A + E + F – A = B – F + F – A which becomes eq.3a) E + F = B – A In eq.5, replace E + F with B – A (from eq.3a): A + C – E = B + B – A which becomes eq.5a) A + C – E = 2×B – A
Hint #2
In eq.4, add A to both sides, and subtract E from both sides: C – A + A – E = A + B + E + A – E which becomes C – E = 2×A + B In eq.5a, replace C – E with 2×A + B: A + 2×A + B = 2×B – A which becomes 3×A + B = 2×B – A In the equation above, add A to both sides, and subtract B from both sides: 3×A + B + A – B = 2×B – A + A – B which makes 4×A = B
Hint #3
In eq.3a, replace B with 4×A: E + F = 4×A – A which becomes eq.3b) E + F = 3×A
Hint #4
Subtract the left and right sides of eq.3 from the left and right sides of eq.5, respectively: A + C – E – (A + E) = B + E + F – (B – F) which is equivalent to A + C – E – A – E = B + E + F – B + F which becomes C – 2×E = E + 2×F Add 2×E to both sides of the above equation: C – 2×E + 2×E = E + 2×F + 2×E which becomes eq.5b) C = 3×E + 2×F
Hint #5
In eq.2, substitute 3×E + 2×F for C (from eq.5b): D = 3×E + 2×F + F which becomes D = 3×E + 3×F which may be written as eq.2a) D = 3×(E + F)
Hint #6
Substitute 3×A for E + F (from eq.3b) in eq.2a: D = 3×(3×A) which makes D = 9×A
Hint #7
eq.6 may be written as: B = (C + E + F) ÷ 3 Substitute 4×A for B, and 3×A for E + F (from eq.3b) in the equation above: 4×A = (C + 3×A) ÷ 3 Multiply both sides by 3: 4×A × 3 = (C + 3×A) ÷ 3 × 3 which becomes 12×A = C + 3×A Subtract 3×A from each side: 12×A – 3×A = C + 3×A – 3×A which makes 9×A = C
Hint #8
Substitute 9×A for D and C in eq.2: 9×A = 9×A + F Subtract 9×A from both sides of the above equation: 9×A – 9×A = 9×A + F – 9×A which makes 0 = F
Hint #9
Substitute 0 for F in eq.3b: E + 0 = 3×A which makes E = 3×A
Solution
Substitute 4×A for B, 9×A for C and D, 3×A for E, and 0 for F in eq.1: A + 4×A + 9×A + 9×A + 3×A + 0 = 26 which simplifies to 26×A = 26 Divide both sides of the equation above by 26: 26×A ÷ 26 = 26 ÷ 26 = 1 which means A = 1 making B = 4×A = 4 × 1 = 4 C = D = 9×A = 9 × 1 = 9 E = 3×A = 3 × 1 = 3 and ABCDEF = 149930