已知五元素数组上下界，求解加速递增的中间值x、y、z

Alright, let's work through this problem clearly. You've got a 5-element array [lower_bound, x, y, z, upper_bound] where the first and last values are fixed, and you need to find x, y, z such that the differences between adjacent elements follow a consistently increasing pattern (your "acceleration" requirement).

First, let's formalize what "accelerating" means here: the gaps between consecutive elements should form an arithmetic sequence—each gap is larger than the previous one by a fixed amount. That's the key to the "consistent increment" rule you mentioned.

Step 1: Define the Differences #

Let’s label the gaps between elements:

• d1 = x - lower_bound (first gap)
• d2 = y - x (second gap)
• d3 = z - y (third gap)
• d4 = upper_bound - z (fourth gap)

For the "accelerating" pattern, each gap increases by a fixed value k (where k > 0 to ensure acceleration):

• d2 = d1 + k
• d3 = d2 + k = d1 + 2k
• d4 = d3 + k = d1 + 3k

Step 2: Relate to Known Bounds #

The total span between the lower and upper bound is the sum of all gaps:

upper_bound - lower_bound = d1 + d2 + d3 + d4

Substitute the gap expressions from above:

upper_bound - lower_bound = d1 + (d1 + k) + (d1 + 2k) + (d1 + 3k)

Simplify the right-hand side:

upper_bound - lower_bound = 4d1 + 6k

Now we have one equation with two variables (d1 and k), which means you can choose one variable based on your needs, then solve for the other.

Step 3: Calculate Middle Values (Two Common Approaches) #

Approach 1: Fix the Initial Gap d1 #

If you know how big you want the first gap to be, solve for k:

k = (upper_bound - lower_bound - 4*d1) / 6

Then compute the middle values:

• x = lower_bound + d1
• y = x + d2 = lower_bound + 2*d1 + k
• z = y + d3 = lower_bound + 3*d1 + 3*k

Approach 2: Fix the Gap Increment k #

If you want to set how much each gap increases by (e.g., each gap is 5 larger than the last), solve for d1:

d1 = (upper_bound - lower_bound - 6*k) / 4

Then compute the middle values using the same formulas as above.

Example Walkthrough #

Let’s say your lower bound is 2 and upper bound is 32:

• If we set d1 = 2, then k = (32-2 - 4*2)/6 = 22/6 ≈ 3.6667
• Calculating middle values:
  • x = 2 + 2 = 4
  • y = 4 + (2 + 3.6667) ≈ 9.6667
  • z = 9.6667 + (2 + 2*3.6667) ≈ 19
• Resulting array: [2, 4, 9.6667, 19, 32] (gaps are 2, 5.6667, 9.3333, 13—each gap increases by ~3.6667)

If we instead set k = 4:

• d1 = (32-2 -6*4)/4 = 6/4 = 1.5
• Middle values:
  • x = 2 +1.5 =3.5
  • y=3.5 + (1.5+4)=9
  • z=9 + (1.5+8)=18.5
• Resulting array: [2, 3.5, 9, 18.5, 32] (gaps are 1.5,5.5,9.5,13.5—each gap increases by exactly 4)

Key Notes #

• Ensure k > 0 to maintain the "accelerating" effect—this means upper_bound - lower_bound must be larger than 4*d1 (if using Approach 1) or 6*k (if using Approach 2).
• If you need integer values, adjust d1 or k so that (upper_bound - lower_bound -4*d1) is divisible by 6, or (upper_bound - lower_bound -6*k) is divisible by 4.

内容的提问来源于stack exchange，提问作者JacobIRR