Puzzle for August 19, 2023 ( )
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.
* "A^C" means "A raised to the power of C".
** "E^A" means "E raised to the power of A".
Scratchpad
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Hint #1
Subtract the left and right sides of eq.3 from the left and right sides of eq.4, respectively: D + E + F - (A + E + F) = A + B + C - E - (B + C + D) which becomes D + E + F - A - E - F = A + B + C - E - B - C - D which becomes D - A = A - E - D Add A, E, and D to both sides of the above equation: D - A + A + E + D = A - E - D + A + E + D which becomes eq.4a) 2×D + E = 2×A
Hint #2
Add A and E to both sides of eq.1: D - A + A + E = A - E + A + E which becomes eq.1a) D + E = 2×A
Hint #3
In eq.1a, replace 2×A with E + 2×D (from eq.4a): D + E = E + 2×D Subtract D and E from each side of the equation above: D + E - D - E = E + 2×D - D - E which means 0 = D
Hint #4
In eq.1a, replace D with 0: 0 + E = 2×A which makes E = 2×A
Hint #5
In eq.2, substitute 2×A for E, and 0 for D: 2×A - 0 = B - 2×A which becomes 2×A = B - 2×A Add 2×A to both sides of the above equation: 2×A + 2×A = B - 2×A + 2×A which makes 4×A = B
Hint #6
Substitute 2×A for E, 4×A for B, and 0 for D in eq.3: A + 2×A + F = 4×A + C + 0 which becomes 3×A + F = 4×A + C Subtract 3×A from each side of the above equation: 3×A + F - 3×A = 4×A + C - 3×A which becomes eq.3a) F = A + C
Hint #7
Substitute 0 for D, (2×A) for E, and 4×A for B in eq.6: A - 0 = ((2×A) ^ A) ÷ 4×A which becomes A = ((2×A) ^ A) ÷ 4×A Multiply both sides of the above equation by 4×A: 4×A × A = 4×A × ((2×A) ^ A) ÷ 4×A which becomes eq.6a) 4×A² = (2×A)^A
Hint #8
Since B ≠ 0 (from eq.6), then A ≠ 0 (since B = 4×A). To make eq.6a true, check several possible values for A > 0: If A = 1, then 4×A² = 4×1² = 4×1 = 4, and (2×A)^A = (2×1)^1 = 2^1 = 2 If A = 2, then 4×A² = 4×2² = 4×4 = 16, and (2×A)^A = (2×2)^2 = 4^2 = 16 If A = 3, then 4×A² = 4×3² = 4×9 = 36, and (2×A)^A = (2×3)^3 = 6^3 = 216 If A = 4, then 4×A² = 4×4² = 4×16 = 64, and (2×A)^A = (2×4)^4 = 8^4 = 4096 If A > 4, the difference between the left and right sides of eq.6a increases. Therefore, the above equations make: A = 2
Hint #9
Since A = 2, then: B = 4×A = 4 × 2 = 8 E = 2×A = 2 × 2 = 4
Hint #10
Substitute 8 for B, 0 for D, and 2 for A in eq.5: 8 + 0 = 2^C which makes 8 = 2^C which means 3 = C
Solution
Substitute 2 for A, and 3 for C in eq.3a: F = 2 + 3 which makes F = 5 and makes ABCDEF = 283045