Puzzle for February 23, 2019 ( )
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.
* B ^ C means "B raised to the power of C".
Scratchpad
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Hint #1
Add (F - B) to both sides of eq.3: E - F + (F - B) = B + (F - B) which becomes eq.3a) E - B = F
Hint #2
Add (C - A) to both sides of eq.4: A - C + F + (C - A) = C + E + (C - A) which becomes eq.4a) F = 2×C + E - A
Hint #3
In eq.4a, replace F with E - B (from eq.3a): E - B = 2×C + E - A Subtract E from each side of the equation above: E - B - E = 2×C + E - A - E which becomes -B = 2×C - A Multiply both sides by (-2): -B × (-2) = (2×C - A) × (-2) which becomes 2×B = -4×C + 2×A which may be written as eq.4b) 2×B = 2×A - 4×C
Hint #4
In eq.5, Substitute (E - B) for F (from eq.3a): D - B = E - (E - B) - A which is equivalent to D - B = E - E + B - A which becomes D - B = B - A Add A + B to both sides of the above equation: D - B + A + B = B - A + A + B which becomes eq.5a) A + D = 2×B
Hint #5
In eq.4b, substitute A + D for 2×B (from eq.5a): A + D = 2×A - 4×C Subtract A from each side of the above equation: A + D - A = 2×A - 4×C - A which simplifies to D = A - 4×C In eq.2, replace D with A - 4×C: A = C + A - 4×C Subtract A from each side: A - A = C + A - 4×C - A which becomes 0 = -3×C which means 0 = C
Hint #6
Substitute 0 for C in eq.2: A = 0 + D which means A = D
Hint #7
Substitute 0 for C in eq.4: A - 0 + F = 0 + E which becomes eq.4c) A + F = E Substitute A + F for E in eq.3: A + F - F = B which means A = B
Hint #8
Substitute 0 for C in eq.6: B ^ 0 = F which makes 1 = F (implies B not = 0)
Hint #9
Substitute 1 for F in eq.4c: eq.4d) A + 1 = E
Solution
Substitute A for B and D, 0 for C, A + 1 for E (from eq.4d), and 1 for F in eq.1: A + A + 0 + A + A + 1 + 1 = 18 which simplifies to 4×A + 2 = 18 Subtract 2 from each side: 4×A + 2 - 2 = 18 - 2 which makes 4×A = 16 Divide both sides by 4: 4×A ÷ 4 = 16 ÷ 4 which means A = 4 making B = D = A = 4 E = A + 1 = 5 (from eq.4d) and ABCDEF = 440451