Puzzle for June 20, 2021 ( )
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: C + F – (C + D) = B + E – (E + F) which is equivalent to C + F – C – D = B + E – E – F which becomes F – D = B – F Add D and F to both sides of the equation above: F – D + D + F = B – F + D + F which becomes eq.3a) 2×F = B + D
Hint #2
Add F to both sides of eq.5: A – F + F = B + F + F which becomes A = B + 2×F In the above equation, replace 2×F with B + D (from eq.3a): A = B + B + D which becomes eq.5a) A = 2×B + D
Hint #3
eq.6 may be written as: B = (A + C + E) ÷ 3 Multiply both sides of the above equation by 3: 3 × B = 3 × (A + C + E) ÷ 3 which becomes eq.6a) 3×B = A + C + E
Hint #4
Add A to both sides of eq.2: B + C + E + A = A + D + A which is equivalent to A + C + E + B = 2×A + D In the above equation, replace A + C + E with 3×B (from eq.6a): 3×B + B = 2×A + D which makes eq.2a) 4×B = 2×A + D
Hint #5
In eq.2a, substitute (2×B + D) for A (from eq.5a): 4×B = 2×(2×B + D) + D which becomes 4×B = 4×B + 2×D + D which becomes 4×B = 4×B + 3×D Subtract 4×B from each side of the above equation: 4×B – 4×B = 4×B + 3×D – 4×B which makes 0 = 3×D which means 0 = D
Hint #6
Substitute 0 for D in eq.5a: A = 2×B + 0 which makes A = 2×B
Hint #7
Substitute 0 for D in eq.3a: 2×F = B + 0 which makes 2×F = B Divide both sides of the equation above by 2: 2×F ÷ 2 = B ÷ 2 which makes F = ½×B
Hint #8
Substitute 2×B for A, and 0 for D in eq.2: B + C + E = 2×B + 0 Subtract B from each side of the above equation: B + C + E – B = 2×B + 0 – B which becomes eq.2b) C + E = B
Hint #9
Substitute 0 for D, and ½×B for F in eq.4: C + 0 = E + ½×B which becomes eq.4a) C = E + ½×B
Hint #10
In eq.2b, substitute E + ½×B for C (from eq.4a): E + ½×B + E = B which becomes ½×B + 2×E = B Subtract ½×B from each side of the above equation: ½×B + 2×E – ½×B = B – ½×B which makes 2×E = ½×B Divide both sides by 2: 2×E ÷ 2 = ½×B ÷ 2 which makes E = ¼×B
Hint #11
Substitute ¼×B for E in eq.4a: C = ¼×B + ½×B which makes C = ¾×B
Solution
Substitute 2×B for A, ¾×B for C, 0 for D, ¼×B for E, and ½×B for F in eq.1: 2×B + B + ¾×B + 0 + ¼×B + ½×B = 18 which simplifies to 4½×B = 18 Divide both sides of the equation above by 4½: 4½×B ÷ 4½ = 18 ÷ 4½ which means B = 4 making A = 2×B = 2 × 4 = 8 C = ¾×B = ¾ × 4 = 3 E = ¼×B = ¼ × 4 = 1 F = ½×B = ½ × 4 = 2 and ABCDEF = 843012