Puzzle for July 17, 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, replace D + F with C + E (from eq.2): B + C + E = A + C – E In the equation above, subtract C from both sides, and add E to both sides: B + C + E – C + E = A + C – E – C + E which becomes eq.3a) B + 2×E = A
Hint #2
In eq.1, replace A with B + 2×E (from eq.3a): B + 2×E = B + E Subtract B and E from each side of the equation above: B + 2×E – B – E = B + E – B – E which means E = 0
Hint #3
In eq.1, substitute 0 for E: A = B + 0 which makes A = B
Hint #4
In eq.3, substitute C + F for B + D (from eq.4), and 0 for E: C + F + F = A + C – 0 which becomes C + 2×F = A + C Subtract C from each side of the equation above: C + 2×F – C = A + C – C which makes 2×F = A and also makes B = A = 2×F
Hint #5
Substitute 2×F for A and B in eq.5: 2×F + 2×F – D = C + D – 2×F which becomes 4×F – D = C + D – 2×F In the above equation, subtract D from both sides, and add 2×F to both sides: 4×F – D – D + 2×F = C + D – 2×F – D + 2×F which becomes eq.5a) 6×F – 2×D = C
Hint #6
Substitute 6×F – 2×D for C (from eq.5a), and 2×F for B in eq.4: 6×F – 2×D + F = 2×F + D which becomes 7×F – 2×D = 2×F + D In the above equation, subtract 2×F from both sides, and add 2×D to both sides: 7×F – 2×D – 2×F + 2×D = 2×F + D – 2×F + 2×D which makes 5×F = 3×D Divide both sides by 3: 5×F ÷ 3 = 3×D ÷ 3 which makes 1⅔×F = D
Hint #7
Substitute (1⅔×F) for D in eq.5a: 6×F – 2×(1⅔×F) = C which becomes 6×F – 3⅓×F = C which makes 2⅔×F = C
Solution
Substitute 2⅔×F for C, 2×F for B and A, and 1⅔×F for D in eq.6: 2⅔×F = ((2×F × 1⅔×F) – 2×F) ÷ F which becomes 2⅔×F = (3⅓×F² – 2×F) ÷ F which becomes 2⅔×F = 3⅓×F – 2 In the above equation, add 2 to both sides, and subtract 2⅔×F from both sides: 2⅔×F + 2 – 2⅔×F = 3⅓×F – 2 + 2 – 2⅔×F which makes 2 = ⅔×F Divide both sides by ⅔: 2 ÷ ⅔ = ⅔×F ÷ ⅔ which makes 3 = F making A = B = 2×F = 2 × 3 = 6 C = 2⅔×F = 2⅔ × 3 = 8 D = 1⅔×F = 1⅔ × 3 = 5 and ABCDEF = 668503