Puzzle for November 12, 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 positive integer.
* "D ^ (F - A)" means "D raised to the power of (F - A)".
Our thanks go out again to Judah S (age 16) for sending us another interesting and challenging puzzle. Thank you, Judah!
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Hint #1
Add C and F to both sides of eq.3: E - C + C + F = C - F + C + F which becomes E + F = 2×C In the above equation, substitute (A + B) for C (from eq.1): E + F = 2×(A + B) which may be written as eq.3a) E + F = 2×A + 2×B
Hint #2
In eq.3a, substitute (A + F) for B (from eq.2): E + F = 2×A + 2×(A + F) which is the same as E + F = 2×A + 2×A + 2×F which becomes E + F = 4×A + 2×F Subtract F from each side of the equation above: E + F - F = 4×A + 2×F - F which becomes eq.3b) E = 4×A + F
Hint #3
In eq.4, substitute A + F for B (from eq.2), and (4×A + F) for E (from eq.3b): A + F + F = A × (4×A + F) which becomes eq.4a) A + 2×F = 4×A² + A×F
Hint #4
Subtract A and A×F from each side of eq.4a: A + 2×F - A - A×F = 4×A² + A×F - A - A×F which becomes 2×F - A×F = 4×A² - A which may be written as F×(2 - A) = 4×A² - A Divide both sides by (2 - A) (assumes A ≠ 2): F×(2 - A) ÷ (2 - A) = (4×A² - A) ÷ (2 - A) which becomes eq.4b) F = (4×A² - A) ÷ (2 - A)
Hint #5
Check: A ≠ 2 ... If A = 2, then substituting 2 for A in eq.4a would yield: 2 + 2×F = 4×2² + 2×F which would become 2 + 2×F = 4×4 + 2×F which would make 2 + 2×F = 16 + 2×F Subtracting 2×F from each side of the above equation would yield: 2 + 2×F - 2×F = 16 + 2×F - 2×F which would make 2 = 16 Since 2 ≠ 16, then: A ≠ 2
Hint #6
To make eq.4b true, check several possible values for A and F: If A = 1, then F = (4×1² - 1) ÷ (2 - 1) = (4 - 1) ÷ 1 = 3 ÷ 1 = 3 If A = 3, then F = (4×3² - 3) ÷ (2 - 3) = (36 - 1) ÷ -1 = -35 If A > 3, then F < -35 Since F is a positive integer, the above equations make: A = 1 and F = 3
Hint #7
In eq.3b, replace A with 1, and F with 3: E = 4×1 + 3 which becomes E = 4 + 3 which makes E = 7
Hint #8
In eq.2, replace A with 1, and F with 3: B = 1 + 3 which makes B = 4
Hint #9
In eq.1, replace A with 1, and B with 4: C = 1 + 4 which makes C = 5
Solution
Substitute 3 for F, 1 for A, 5 for C, and 7 for E in eq.5: D ^ (3 - 1) = 1 + (5 × 7) which becomes D ^ 2 = 1 + 35 which makes D² = 36 which means D = 6 and makes ABCDEF = 145673