Puzzle for April 23, 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 positive integer.
Scratchpad
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Hint #1
Add E to both sides of eq.4: C + D + E = A + B – E + E which becomes C + D + E = A + B In the above equation, replace D + E with B (from eq.3): C + B = A + B Subtract B from each side: C + B – B = A + B – B which makes C = A
Hint #2
In eq.5, replace F with A + E (from eq.2): D + A + E = A + C Subtract A from each side of the equation above: D + A + E – A = A + C – A which becomes eq.5a) D + E = C
Hint #3
In eq.3, substitute C for D + E (from eq.5a): B = C which makes A = B = C
Hint #4
In eq.3, substitute A for B: A = D + E Subtract E from both sides of the equation above: A – E = D + E – E which becomes eq.3a) A – E = D
Hint #5
In eq.6, substitute A for B, (A – E) for D (from eq.3a), and (A + E) for F (from eq.2): A × A = ((A – E) × (A + E)) + E which becomes A² = (A² – E×A + E×A – E²) + E which becomes A² = A² – E² + E In the equation above, subtract A² from both sides, and add E² to both sides: A² – A² + E² = A² – E² + E – A² + E² which simplifies to E² = E Divide both sides by E: E² ÷ E = E ÷ E which makes E = 1
Hint #6
Substitute A for C in eq.5: D + F = A + A which becomes eq.5a) D + F = 2×A
Solution
eq.1 may be written as: A + B + C + D + F + E = 41 Substitute A for B and C, 2×A for D + F (from eq.5a), and 1 for E in the above equation: A + A + A + 2×A + 1 = 41 which becomes 5×A + 1 = 41 Subtract 1 from both sides of the above equation: 5×A + 1 – 1 = 41 – 1 which makes 5×A = 40 Divide both sides by 5: 5×A ÷ 5 = 40 ÷ 5 which means A = 8 and makes B = C = A = 8 and also makes D = A – E = 8 – 1 = 7 (from eq.3a) F = A + E = 8 + 1 = 9 (from eq.2) and ABCDEF = 888719