Puzzle for July 16, 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
Subtract the left and right sides of eq.4 from the left and right sides of eq.3, respectively: A + B – (B + E) = C + D – (A + C) which becomes A + B – B – E = C + D – A – C which becomes A – E = D – A Add E and A to both sides of the above equation: A – E + E + A = D – A + E + A which becomes eq.3a) 2×A = D + E
Hint #2
Subtract the left and right sides of eq.2 from the left and right sides of eq.3, respectively: A + B – (B + D) = C + D – (C + E) which becomes A + B – B – D = C + D – C – E which becomes A – D = D – E Add D to both sides of the equation above: A – D + D = D – E + D which becomes eq.3b) A = 2×D – E
Hint #3
In eq.3a, substitute (2×D – E) for A (from eq.3b): 2×(2×D – E) = D + E which becomes 4×D – 2×E = D + E In the above equation, add 2×E to both sides, and subtract D from both sides: 4×D – 2×E + 2×E – D = D + E + 2×E – D which becomes 3×D = 3×E Divide both sides by 3: 3×D ÷ 3 = 3×E ÷ 3 which makes D = E
Hint #4
In eq.3b, replace E with D: A = 2×D – D which makes A = D
Hint #5
In eq.4, replace E and A with D: B + D = D + C Subtract D from each side of the equation above: B + D – D = D + C – D which makes B = C
Hint #6
In eq.5, substitute A for E: A + F = C + A – F In the above equation, subtract A from both sides, add F to both sides: A + F – A + F = C + A – F – A + F which makes 2×F = C and also makes 2×F = C = B
Hint #7
eq.6 may be written as: A – C + E – F = (B + F) ÷ 2 Multiply both sides of the above equation by 2: 2 × (A – C + E – F) = 2 × (B + F) ÷ 2 which becomes 2×A – 2×C + 2×E – 2×F = B + F Add 2×C and 2×F to both sides: 2×A – 2×C + 2×E + 2×C – 2×F + 2×F = B + F + 2×C + 2×F which becomes eq.6a) 2×A + 2×E = B + 3×F + 2×C
Hint #8
Substitute A for E, and (2×F) for B and C in eq.6a: 2×A + 2×A = (2×F) + 3×F + 2×(2×F) which becomes 4×A = 5×F + 4×F which makes 4×A = 9×F Divide both sides by 4: 4×A ÷ 4 = 9×F ÷ 4 which makes A = 2¼×F and also makes E = D = A = 2¼×F
Solution
Substitute 2¼×F for A and D and E, and 2×F for B and C in eq.1: 2¼×F + 2×F + 2×F + 2¼×F + 2¼×F + F = 47 which simplifies to 11¾×F = 47 Divide both sides of the above equation by 11¾: 11¾×F ÷ 11¾ = 47 ÷ 11¾ which means F = 4 making A = D = E = 2¼×F = 2¼ × 4 = 9 B = C = 2×F = 2 × 4 = 8 and ABCDEF = 988994