Puzzle for June 4, 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 F with D + E (from eq.6): C + E = B + D + D + E Subtract E from both sides of the above equation: C + E – E = B + D + D + E – E which becomes eq.3a) C = B + 2×D
Hint #2
In eq.3a, replace B with C + D (from eq.5): C = C + D + 2×D which becomes C = C + 3×D Subtract C from each side of the equation above: C – C = C + 3×D – C which makes 0 = 3×D which means 0 = D
Hint #3
In eq.5, substitute 0 for D: B = C + 0 which makes B = C
Hint #4
In eq.6, substitute 0 for D: 0 + E = F which makes E = F
Hint #5
Substitute B for C, and 0 for D in eq.4: F – (B + B) = A + B + B – 0 which becomes F – 2×B = A + 2×B Add 2×B to each side of the above equation: F – 2×B + 2×B = A + 2×B + 2×B which becomes eq.4a) F = A + 4×B
Hint #6
Add A to both sides of eq.2: E – A + A = A – B + A which becomes E = 2×A – B Substitute F for E in the equation above: eq.2a) F = 2×A – B
Hint #7
Substitute 2×A – B for F (from eq.2a) in eq.4a: 2×A – B = A + 4×B In the equation above, add B to each side, and subtract A from each side: 2×A – B + B – A = A + 4×B + B – A which makes A = 5×B
Hint #8
Substitute 5×B for A in eq.4a: F = 5×B + 4×B which makes F = 9×B and which also makes E = F = 9×B
Solution
Substitute 5×B for A, B for C, 0 for D, and 9×B for E and F in eq.1: 5×B + B + B + 0 + 9×B + 9×B = 25 which simplifies to 25×B = 25 Divide each side of the equation above by 25: 25×B ÷ 25 = 25 ÷ 25 which means B = 1 making A = 5×B = 5 × 1 = 5 C = B = 1 E = F = 9×B = 9 × 1 = 9 and ABCDEF = 511099