Puzzle for August 6, 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
eq.6 may be written as: E – B = (C + D) ÷ 2 Multiply both sides of the above equation by 2: 2 × (E – B) = 2 × (C + D) ÷ 2 which becomes eq.6a) 2×E – 2×B = C + D Add D to both sides of eq.3: C – D + D = B + D + D which becomes eq.3a) C = B + 2×D
Hint #2
In eq.6a, replace C with B + 2×D (from eq.3a): 2×E – 2×B = B + 2×D + D which becomes 2×E – 2×B = B + 3×D Add 2×B to both sides of the equation above: 2×E – 2×B + 2×B = B + 3×D + 2×B which becomes eq.6b) 2×E = 3×B + 3×D
Hint #3
Subtract E from each side of eq.4: B + D – E = A + E – F – E which becomes eq.4a) B + D – E = A – F Subtract F and B from both sides of eq.2: F – B – F – B = A + B – F – B which becomes eq.2a) –2×B = A – F
Hint #4
In eq.2a, replace A – F with B + D – E (from eq.4a): –2×B = B + D – E Subtract B and D from both sides of the above equation: –2×B – B – D = B + D – E – B – D which becomes –3×B – D = –E Multiply both sides by (–1): (–1) × (–3×B – D) = (–1) × (–E) which becomes eq.2b) 3×B + D = E
Hint #5
In eq.6b, substitute (3×B + D) for E (from eq.2b): 2×(3×B + D) = 3×B + 3×D which becomes 6×B + 2×D = 3×B + 3×D Subtract 2×D and 3×B from each side of the equation above: 6×B + 2×D – 2×D – 3×B = 3×B + 3×D – 2×D – 3×B which simplifies to 3×B = D
Hint #6
Substitute 3×B for D into eq.3a: C = B + 2×(3×B) which becomes C = B + 6×B which makes C = 7×B
Hint #7
Substitute 3×B for D in eq.2b: 3×B + 3×B = E which makes 6×B = E
Hint #8
Substitute 3×B for D, 6×B for E, and 7×B for C in eq.5: 3×B + 6×B = A – 7×B + F which becomes 9×B = A – 7×B + F Add 7×B to both sides of the equation above: 9×B + 7×B = A – 7×B + F + 7×B which becomes eq.5a) 16×B = A + F
Hint #9
Add the left and right sides of eq.2a to the left and right sides of eq.5a, respectively: 16×B + (–2×B) = A + F + A – F which makes 14×B = 2×A Divide both sides of the above equation by 2: 14×B ÷ 2 = 2×A ÷ 2 which makes 7×B = A
Hint #10
Substitute 7×B for A in eq.2: F – B = 7×B + B which becomes F – B = 8×B Add B to both sides of the equation above: F – B + B = 8×B + B which makes F = 9×B
Solution
Substitute 7×B for A and C, 3×B for D, 6×B for E, and 9×B for F in eq.1: 7×B + B + 7×B + 3×B + 6×B + 9×B = 33 which simplifies to 33×B = 33 Divide both sides of the above equation by 33: 33×B ÷ 33 = 33 ÷ 33 which means B = 1 making A = C = 7×B = 7 × 1 = 7 D = 3×B = 3 × 1 = 3 E = 6×B = 6 × 1 = 6 F = 9×B = 9 × 1 = 9 and ABCDEF = 717369