# Distribute in Circle | InterviewBit | Solution Explained

I explain the solution to Distribute in Circle InterviewBit problem. Using visuals and intuitions for the mathematical solution. Problem Link.

## 0. Distribute in Circle: Problem Discussion

### 0.0. IO Format

- input:
- A: number of items
- B: size of circle
- C: starting point inside the circle

- output:
- position where the Ath item (last) will get delivered

- constraints:
- 1 <= A, B, C <= 10
^{8}

- 1 <= A, B, C <= 10

### 0.1. Examples

In this case, we have

- a circle of B size – marked in blue.
- inside of the circle, we first move to the Cth position (zero indexed)
- then start distributing A items from that position

In this case, we start from index C = 4, then go in the clockwise direction in the circle B = 5 and distribute A = 7 items. The last item is distributed at the 0th index, which is thus the answer.

## 1. Distribute in Circle: Observations & Logic

### 1.0. Intuitions

One solid point to notice is that we are stuck inside a circle, so the Ath item only has an effect of A % B. What it means is, if we have A = 7 and B = 5, we basically will allocate A / B = 1 items to everyone, then the next A % B = 2 people will get one more item than the rest. NOTE: the minus 1 is done because we will be at A % B position when we have successfully distributed everything. That means that the last index that we actually distributed was actually A % B – 1.

**formula = A % B – 1**

Now where does C come into play? Think of it as an initial displacement – we could have started from 0th index, and returned the answer as A % B – 1. C just comes in, and displaces everything by C amount.

**formula = (C + A % B – 1) % B**

NOTE: we have again done % B since the value inside the brackets can actually go beyond B. Think why!

## 2. Distribute in Circle: Optimized Implementation

### 2.0. Code

int Solution::solve(int A, int B, int C) { return (C + A % B - 1) % B; }

### 2.1. Complexity Analysis

- Time:
`O(1)`

, since we only do O(1) operations - Space:
`O(1)`

, since we never use any variable ourselves.

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Please write the proof.

This solution need an amendment for 2 digit number.