Problem:
Score: 300 points

Problem Statement

In the Ancient Kingdom of Snuke, there was a pyramid to strengthen the authority of Takahashi, the president of AtCoder Inc.

The pyramid had center coordinates (C_X, C_Y) and height H. The altitude of coordinates (X, Y) is max(H - |X - C_X| - |Y - C_Y|, 0).

Aoki, an explorer, conducted a survey to identify the center coordinates and height of this pyramid. As a result, he obtained the following information:

C_X, C_Y was integers between 0 and 100 (inclusive), and H was an integer not less than 1.

Additionally, he obtained N pieces of information. The i-th of them is: "the altitude of point (x_i, y_i) is h_i."

This was enough to identify the center coordinates and the height of the pyramid. Find these values with the clues above.

Constraints

N is an integer between 1 and 100 (inclusive).

x_i and y_i are integers between 0 and 100 (inclusive).

h_i is an integer between 0 and 10^9 (inclusive).

The N coordinates (x_1, y_1), (x_2, y_2), (x_3, y_3), ..., (x_N, y_N) are all different.

The center coordinates and the height of the pyramid can be uniquely identified.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1 h_1
x_2 y_2 h_2
x_3 y_3 h_3
:
x_N y_N h_N

Output

Print values C_X, C_Y and H representing the center coordinates and the height of the pyramid in one line, with spaces in between.

Sample Input 1

4
2 3 5
2 1 5
1 2 5
3 2 5

Sample Output 1

2 2 6

In this case, the center coordinates and the height can be identified as (2, 2) and 6.

Sample Input 2

2
0 0 100
1 1 98

Sample Output 2

0 0 100

In this case, the center coordinates and the height can be identified as (0, 0) and 100.

Note that C_X and C_Y are known to be integers between 0 and 100.

Sample Input 3

3
99 1 191
100 1 192
99 0 192

Sample Output 3

100 0 193

In this case, the center coordinates and the height can be identified as (100, 0) and 193.

Solution in Kotlin (1.0.0) (judge verdict: Accepted)

import java.io.PrintWriter
import java.util.*
import java.math.*
import java.lang.*
import kotlin.comparisons.*

val pw = PrintWriter(System.out)
val MOD: Long = (1e+9 + 7).toLong()

data class Point(val x: Int, val y: Int, val h: Int)

fun main(args: Array<String>) {
    val n = readInt()        
    val points = (0 until n).map {
        val (x, y, h) = readListOfInt()
        Point(x, y, h)
    }.sortedBy { -it.h }

    for(cx in 0..100) {
        for(cy in 0..100) {
            val pt = points[0]
            val ch = pt.h + Math.abs( pt.x - cx ) + Math.abs( pt.y - cy )
            if(points.all { p -> p.h ==  Math.max( ch - Math.abs(p.x - cx) - Math.abs(p.y - cy), 0) }) {
                println("$cx $cy $ch")
                pw.flush()
                return
            }
        }
    }

    pw.flush()
}

/****** Declared Functions and Data Structures ******/
// IO
fun read() = readLine()!!
fun readInt() = read().toInt()
fun readLong() = read().toLong()
fun readDouble() = read().toDouble()
fun readListOfString() = read().split(" ")
fun readListOfInt() = readListOfString().map { it.toInt() }
fun readListOfLong() = readListOfString().map { it.toLong() }
fun readListOfDouble() = readListOfString().map { it.toDouble() }

fun Double.format(digits: Int) = String.format("%.${digits}f", this)
fun Float.format(digits: Int) = String.format("%.${digits}f", this)

fun print(value: Any) { pw.print(value) }
fun println(value : Any) { pw.println(value) }

fun<T> permutations(src: List<T>): List<List<T>> {
    if(src.size == 1) return listOf(src)
    val perms = mutableListOf<List<T>>()
    val insertElement = src[0]
    permutations(src.drop(1)).forEach { perm ->
        for(i in 0..perm.size) {
            val newPerm = perm.toMutableList()
            newPerm.add(i, insertElement)
            perms.add(newPerm.toList())
        }
    }
    return perms
}

// return nCr, if you want nCr, please access v[n][r]
fun combinations(n: Long): List<List<Long>> {
    val v = (0..n).map { (0..n).map{0L}.toMutableList() }.toMutableList()
    for(i in 0 until v.size) {
        v[i][0] = 1L
        v[i][i] = 1L
    }
    for(i in 0 until v.size) {
        for(j in 1 until i) {
            v[i][j] = (v[i-1][j-1] + v[i-1][j]) % MOD
        }
    }
    return v.map { it.toList() }.toList()
}

// should be a >= b
fun gcd(a: Long, b: Long): Long = 
    if(b == 0L) a else gcd(b, a % b)

// shoud be a >= b
fun lcm(a: Long, b: Long): Long = 
    a / gcd(a, b) * b

fun sumDigits(num_: Long): Long {
    var num = num_
    var rtn: Long = 0
    while(num != 0.toLong()) {
        val tmp = num % 10
        rtn += tmp
        num /= 10
    }
    return rtn
}

fun isDecimal(v: Double): Boolean = ((v - Math.floor(v)) != 0.0)

// Data Structures
class Stack<T>(private var st: MutableList<T> = mutableListOf<T>()) {
    fun isEmpty() = st.isEmpty()
    fun top(): T = st.last()
    fun push(e: T) { st.add(e) }
    fun pop() { st = st.dropLast(1).toMutableList() }
}

class Queue<T>(private var que: MutableList<T> = mutableListOf<T>()) {
    fun isEmpty() = que.isEmpty()
    fun top(): T = que.first()
    fun push(e: T) { que.add(e) }
    fun pop() { que = que.drop(1).toMutableList() }
}