Assessing the problem
Identifying skill data
Establishing skill constraints
Evaluating skill combinations
Assessing skill line options
Determining skill line options
Determining skill line constraints
Calculating group skill constraints
Calculating skill constraints
Ensuring skill thresholds
Mapping out skill constraints
I’m calculating the maximum values for each skill in the group. Summing these, I found the total skill cap is 120.
Filling in gaps
OK, let me see. Colonist #4 needs to bump the skill total from 120 to 140, which means a combined 20 increment across all skills.
Mapping out practical limits
I’m noting that each skill’s increment is capped by the possible increase from its current maximum to a theoretical cap of 20.
Breaking down the constraint
I'm noting that the increments for each skill need to be nonnegative and within set limits, totaling 20. This approach simplifies managing skill growth.
Calculating potential gains
I’m working out the possible increments for each skill, ensuring they match the constraints and total 20. This helps in defining the feasible range for each skill level.
Piecing it together
I’m thinking through valid skill increments: selecting y_i > 0 balances with x_i ≤ c_i, controlling combinations while ensuring the sum of increments equals 20.
Weighing solutions
I’m mapping out the total solutions by tallying the product of choices for each y_i, factoring in its constraints and ensuring the sum of y_i equals 20.
Breaking it down
I’m detailing the method to count solutions by partitioning 20 among up to 12 "bins" with specific capacities, ensuring each bin's choices align with its constraints.
Breaking down the problem
I’m examining a combinational formula to simplify the calculation of solutions, ensuring each bin's choices align with constraints and summing the product of factors for valid solutions.
Crafting polynomial strategies
I’m piecing together polynomial strategies to handle skill levels and their constraints, aiming to capture the balance between y_i=0 and y_i>0, and their influence on the total sum.
Building upon generating functions
I’m constructing generating functions for each skill to capture valid solutions. The coefficient of t^{20} in the product of these functions reveals the total number of valid solution sets.
Balancing the functions
I’m working on combining terms so that the exponent of t aligns with the total sum of skill levels, ensuring all valid solution sets are included for t^{20}.
Formulating the approach
I’m defining generating functions for each skill, ensuring the sum of powers form captures valid solution sets. Evaluating the coefficient of t^{20} in the product reveals the total valid solutions.
Mapping out dynamic programming
I’m setting up a table to count ways to form t^{20} using skill generating functions, considering coefficients and exponents to systematically capture all valid solutions.
Mapping coefficients
I’m carefully defining coefficients and exponents for each generating function, ensuring they reflect the valid solution sets for the specific skill levels.
Building out the DP table
I’m constructing the DP array systematically, updating it for each skill index and sum up to 20. The goal is to pinpoint the coefficient of t^{20}.
Crafting the DP array
I’m extending the DP table with new coefficients. For sums up to 14, DP[i][s] updates to 1 for each, and remains 0 for sums beyond 14.
Piecing together DP values
I’m calculating DP values for sums 0 to 20, using previous results and coefficients systematically. For sums up to 14, they update to 1; beyond 14, they remain 0.
Piecing together DP values
I’m meticulously summing valid previous DP values for each skill level, ensuring sums up to 14 update to 1 and remain 0 beyond 14.
Calculating cumulative DP values
I’m updating DP values for each skill level, ensuring sums remain accurate. For skill levels up to 14, DP values are 1; for levels above 14, they are 0.
Mapping out contours
I'm assembling contours to be filled in the DP array, ensuring each sum up to 14 has a value of 1, while sums beyond 14 retain a value of 0.
Laying out options
Focusing on skill levels 15 to 20, I'm curious about the total number of ways considering caps and only one potential casting reversal. Summing combinations gives the result.
Piecing together DP values
OK, let's tweak the DP values for sums 15 to 20 using the derived coefficients and previous sums. The coefficients are carefully mapped for accuracy.
Computing partial sums
Using the S2 array to streamline cumulative sums for DP[2][s]. This method makes calculating DP[3][s] more efficient and subsequent calculations more manageable.
Calculating cumulative sums
I’m working through computing DP[3][s] for s values up to 20 using the partial sums method. For s above 14, subtracting sums as specified helps maintain accuracy.
Mapping out DP values
OK, I’ve pinpointed DP[3][s] values for sums up to 20 using the partial sums method. Now, defining DP[4][s] involves integrating these sums with coefficients and previous values.
Piecing together DP values
I’m assembling DP[4][s] values by leveraging the S3 array and using coefficients with partial sums for efficiency. This approach ensures precision and smooth computation up to s=20.
Breaking down DP calculations
Systematically calculating DP[4][s] for sums 1 to 20 using the S3 array. Partial sums are applied for precision, confirming correct results up to s=20.
Crunching DP values
I’m calculating DP[5][s] using the S4 array, applying coefficients and partial sums for efficiency. This approach ensures precision, covering sums up to 20.
Mapping DP values
Recalculating DP[5][s] using the S4 array, coefficients, and partial sums. Ensuring values up to s=20 are accurate, always confirming precision and consistency throughout.
Noticing patterns
I calculated S5 for sums up to 20. This involved partial summation and coefficients from the previous layer. Now, setting up DP[6][s] for better efficiency.
Crunching DP values
I calculated DP[6][s] using the S5 array, incorporating coefficients and partial sums. This determination spans sums up to 20, ensuring precision throughout.
Crunching DP values
I’m thinking through DP[7][s] using the S6 array, applying coefficients and partial sums for efficiency. This ensures the precision of values up to s=20.
Calculating DP values
I’m piecing together DP[7][s] using the S6 array, coefficients, and partial sums. This method ensures precision for sums up to s=20.
Crunching sums and updating DP
I’m ensuring precise calculations for DP[8][s] using the S7 array, coefficients, and partial sums. Valued sums up to s=20 are accurately determined.
Piecing together DP values
I calculated DP[8][s] with S7, coefficients, and partial sums. This method ensures accuracy for sums up to s=20, streamlining the process significantly.
Calculating and summing
I’m pulling together DP[8][s] using S7, coefficients, and partial sums. This approach ensures precise calculations for sums up to s=20.
Ensuring accurate computation
OK, let me see. I’m carefully computing DP values up to s=20 using S8, coefficients, and partial sums. This guarantees precise and consistent values.
Crafting DP values
I’m building DP[9][s] with S8, coefficients, and partial sums. Pinpoint accuracy for sums up to s=20 is achieved.ימו
Handling sum calculations
I’m carefully calculating DP values up to s=20 using S9, coefficients, and partial sums. This ensures precision throughout each step.
Mapping out DP[10][s]
I’m using S9, coefficients, and partial sums to pinpoint DP values up to s=20. This leads to precise calculations, ensuring accurate progress in the process.
Assessing contributions
OK, let me see. Considering coefficients, partial sums, and sums of earlier DP values, I’m calculating DP values up to s=20 for precise contributions.
Assembling DP values
I'm formulating DP values up to s=20 by applying coefficients, partial sums, and earlier DP values, ensuring each calculation is thorough and precise.
Estimating contributions
I'm calculating DP values up to s=20 by tweaking coefficients, partial sums, and sums of previous DP values, aiming for precision and thoroughness.
Ensuring precision
I'm recalculating DP values up to s=20, carefully checking each digit to confirm accuracy. This involves refining coefficients, partial sums, and previous DP values to guarantee precise results.
Ensuring accurate sums
I'm adding large numbers carefully, using column addition and partial sums to avoid mistakes. This method helps confirm precise DP values up to s=20.
Examining calculations
I'm piecing together partial sums and column addition to ensure the final result is accurate. Thankfully, this careful approach uses scientific notation for clarity.
Confirming the total
I’m checking the final sum by converting large numbers to scientific notation, then back to integers, to ensure we didn’t miss any digits. The sum is confirmed as 144,829,262,996.
Ensuring accuracy
Piecing together DP values up to s=16, I’m aligning partial sums with previous values. This systematic approach is refining the final contributions.
Verifying calculations
Taking a closer look at the multiplication step, ensuring the factor of 16 is accounted for as 10 plus 6, leading to a sum of 179,597,254,704.
Calculating and verifying
I’m finalizing DP values up to s=17, ensuring partial sums and direct calculations. It’s interesting to see the alignment checks confirm the final sum as 217,902,812,874.
Breaking down calculations
I’m carefully calculating DP values up to s=20, ensure partial sums and direct multiplications align. The final sum is 327,634,800,883.
Breaking down the formula
I’m tracking DP values up to s=3 for DP[11][s], using partial sums and previous DP values efficiently to streamline the next steps.
Calculating and summing
I’m working through DP[11][s] for s=5 to s=20, effectively using partial sums and previous DP values to ensure accurate results. Progressing through, each step solidifies the calculations.
Mapping out the final steps
I’m calculating and summing DP[11][s] values up to s=20, using partial sums and previous DP values. The final result is 6,604,511,793,373, ensuring accuracy throughout.
Calculating and summing
I’m working through DP[11][s] values up to s=20, using partial sums and previous DP values. The final sum for S11(s) is approximately 49 trillion.
Refining the formula
I’m adjusting the formula for DP[12][s] by excluding sums up to DP[11][12]. This ensures accuracy for DP[12][20], as each step is carefully refined to avoid errors.
Summing it carefully
I’m working through DP[12][20] by multiplying DP[11][20] by 14 and adding the sum of differences between partial sums. The final result for DP[12][20] is 123,128,800,226,334.
Assessing the count
I’m thinking the final tally of 123,128,800,226,334 possible 12-tuples aligns with the problem's constraints. This large number suggests a significant range of skill combinations for colonist #4.
Reasoned about settlers' skill set and group selection for 8m 53s