Puzzle for July 9, 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
Add F to both sides of eq.4: B + F + F = C – F + F which becomes eq.4a) B + 2×F = C Add A to both sides of eq.5: C – A + A = A + B + E + A which becomes eq.5a) C = 2×A + B + E
Hint #2
In eq.4a, replace C with 2×A + B + E (from eq.5a): B + 2×F = 2×A + B + E Subtract 2×A and B from each side of the equation above: B + 2×F – 2×A – B = 2×A + B + E – 2×A – B which becomes eq.4b) 2×F – 2×A = E
Hint #3
In eq.2, replace E with 2×F – 2×A (from eq.4b): B + 2×F – 2×A = A Add 2×A to both sides of the equation above: B + 2×F – 2×A + 2×A = A + 2×A which becomes eq.2a) B + 2×F = 3×A
Hint #4
In eq.3, substitute 2×F – 2×A for E (from eq.4b): F – B = B + 2×F – 2×A In the equation above, add 2×A to both sides, and subtract B and F from both sides: F – B + 2×A – B – F = B + 2×F – 2×A + 2×A – B – F which becomes eq.3a) 2×A – 2×B = F
Hint #5
Substitute (2×A – 2×B) for F (from eq.3a) into eq.2a: B + 2×(2×A – 2×B) = 3×A which becomes B + 4×A – 4×B = 3×A which becomes 4×A – 3×B = 3×A In the above equation, add 3×B to both sides, and subtract 3×A from both sides: 4×A – 3×B + 3×B – 3×A = 3×A + 3×B – 3×A which becomes A = 3×B
Hint #6
Substitute (3×B) for A in eq.3a: 2×(3×B) – 2×B = F which becomes 6×B – 2×B = F which makes 4×B = F
Hint #7
Substitute 3×B for A in eq.2: B + E = 3×B Subtract B from each side of the above equation: B + E – B = 3×B – B which makes E = 2×B
Hint #8
Substitute 4×B for F in eq.4a: B + 2×(4×B) = C which becomes B + 8×B = C which makes 9×B = C
Hint #9
Substitute 3×B for A, 2×B for E, 4×B for F, and 9×B for C in eq.6: 3×B – D + 2×B + 4×B = 9×B + D which becomes 9×B – D = 9×B + D In the above equation, add D to both sides, and subtract 9×B from both sides: 9×B – D + D – 9×B = 9×B + D + D – 9×B which makes 0 = 2×D which means 0 = D
Solution
Substitute 3×B for A, 9×B for C, 0 for D, 2×B for E, and 4×B for F in eq.1: 3×B + B + 9×B + 0 + 2×B + 4×B = 19 which simplifies to 19×B = 19 Divide both sides of the above equation by 19: 19×B ÷ 19 = 19 ÷ 19 which means B = 1 making A = 3×B = 3 × 1 = 3 C = 9×B = 9 × 1 = 9 E = 2×B = 2 × 1 = 2 F = 4×B = 4 × 1 = 4 and ABCDEF = 319024