Problem:
Score : 200 points

Problem Statement

There are N points in a D-dimensional space.

The coordinates of the i-th point are (X_{i1}, X_{i2}, ..., X_{iD}).

The distance between two points with coordinates (y_1, y_2, ..., y_D) and (z_1, z_2, ..., z_D) is \sqrt{(y_1 - z_1)^2 + (y_2 - z_2)^2 + ... + (y_D - z_D)^2}.

How many pairs (i, j) (i < j) are there such that the distance between the i-th point and the j-th point is an integer?

Constraints

All values in input are integers.

2 \leq N \leq 10

1 \leq D \leq 10

-20 \leq X_{ij} \leq 20

No two given points have the same coordinates. That is, if i \neq j, there exists k such that X_{ik} \neq X_{jk}.

Input

Input is given from Standard Input in the following format:

N D
X_{11} X_{12} ... X_{1D}
X_{21} X_{22} ... X_{2D}
\vdots
X_{N1} X_{N2} ... X_{ND}

Output

Print the number of pairs (i, j) (i < j) such that the distance between the i-th point and the j-th point is an integer.

Sample Input 1

3 2
1 2
5 5
-2 8

Sample Output 1

1

The number of pairs with an integer distance is one, as follows:

The distance between the first point and the second point is \sqrt{|1-5|^2 + |2-5|^2} = 5, which is an integer.

The distance between the second point and the third point is \sqrt{|5-(-2)|^2 + |5-8|^2} = \sqrt{58}, which is not an integer.

The distance between the third point and the first point is \sqrt{|-2-1|^2+|8-2|^2} = 3\sqrt{5}, which is not an integer.

Sample Input 2

3 4
-3 7 8 2
-12 1 10 2
-2 8 9 3

Sample Output 2

2

Sample Input 3

5 1
1
2
3
4
5

Sample Output 3

10

Solution in Rust (1.42.0) (judge verdict: Accepted)

use lazy_static::lazy_static;
use proconio::{fastout, input};
use std::collections::BTreeSet;

lazy_static! {
    static ref SQUARES: BTreeSet<i32> = [
        1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400,
        441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369,
        1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704,
        2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4096, 4225, 4356, 4489,
        4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724,
        6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409,
        9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025, 11236, 11449, 11664, 11881, 12100,
        12321, 12544, 12769, 12996, 13225, 13456, 13689, 13924, 14161, 14400, 14641, 14884, 15129,
        15376, 15625, 15876, 16129, 16384, 16641, 16900, 17161, 17424, 17689, 17956, 18225, 18496,
        18769, 19044, 19321, 19600, 19881, 20164, 20449, 20736, 21025, 21316, 21609, 21904, 22201,
        22500, 22801, 23104, 23409, 23716, 24025, 24336, 24649, 24964, 25281, 25600, 25921, 26244,
        26569, 26896, 27225, 27556, 27889, 28224, 28561, 28900, 29241, 29584, 29929, 30276, 30625,
        30976, 31329, 31684, 32041, 32400, 32761, 33124, 33489, 33856, 34225, 34596, 34969, 35344,
        35721, 36100, 36481, 36864, 37249, 37636, 38025, 38416, 38809, 39204, 39601, 40000, 40401,
        40804, 41209, 41616, 42025, 42436, 42849, 43264, 43681, 44100, 44521, 44944, 45369, 45796,
        46225, 46656, 47089, 47524, 47961, 48400, 48841, 49284, 49729, 50176, 50625, 51076, 51529,
        51984, 52441, 52900, 53361, 53824, 54289, 54756, 55225, 55696, 56169, 56644, 57121, 57600,
        58081, 58564, 59049, 59536, 60025, 60516, 61009, 61504, 62001, 62500, 63001, 63504, 64009,
        64516, 65025, 65536, 66049, 66564, 67081, 67600, 68121, 68644, 69169, 69696, 70225, 70756,
        71289, 71824, 72361, 72900, 73441, 73984, 74529, 75076, 75625, 76176, 76729, 77284, 77841,
        78400, 78961, 79524, 80089, 80656, 81225, 81796, 82369, 82944, 83521, 84100, 84681, 85264,
        85849, 86436, 87025, 87616, 88209, 88804, 89401, 90000, 90601, 91204, 91809, 92416, 93025,
        93636, 94249, 94864, 95481, 96100, 96721, 97344, 97969, 98596, 99225, 99856, 100489,
        101124, 101761, 102400, 103041, 103684, 104329, 104976, 105625, 106276, 106929, 107584,
        108241, 108900, 109561, 110224, 110889, 111556, 112225, 112896, 113569, 114244, 114921,
        115600, 116281, 116964, 117649, 118336, 119025, 119716, 120409, 121104, 121801, 122500,
        123201, 123904, 124609, 125316, 126025, 126736, 127449, 128164, 128881, 129600, 130321,
        131044, 131769, 132496, 133225, 133956, 134689, 135424, 136161, 136900, 137641, 138384,
        139129, 139876, 140625, 141376, 142129, 142884, 143641, 144400, 145161, 145924, 146689,
        147456, 148225, 148996, 149769, 150544, 151321, 152100, 152881, 153664, 154449, 155236,
        156025, 156816, 157609, 158404, 159201, 160000, 160801, 161604, 162409, 163216, 164025,
        164836, 165649, 166464, 167281, 168100, 168921, 169744, 170569, 171396, 172225, 173056,
        173889, 174724, 175561, 176400, 177241, 178084, 178929, 179776, 180625, 181476, 182329,
        183184, 184041, 184900, 185761, 186624, 187489, 188356, 189225, 190096, 190969, 191844,
        192721, 193600, 194481, 195364, 196249, 197136, 198025, 198916, 199809, 200704, 201601,
        202500, 203401, 204304, 205209, 206116, 207025, 207936, 208849, 209764, 210681, 211600,
        212521, 213444, 214369, 215296, 216225, 217156, 218089, 219024, 219961, 220900, 221841,
        222784, 223729, 224676, 225625, 226576, 227529, 228484, 229441, 230400, 231361, 232324,
        233289, 234256, 235225, 236196, 237169, 238144, 239121, 240100, 241081, 242064, 243049,
        244036, 245025, 246016, 247009, 248004, 249001, 250000, 251001, 252004, 253009, 254016,
        255025, 256036, 257049, 258064, 259081, 260100, 261121, 262144, 263169, 264196, 265225,
        266256, 267289, 268324, 269361, 270400, 271441, 272484, 273529, 274576, 275625, 276676,
        277729, 278784, 279841, 280900, 281961, 283024, 284089, 285156, 286225, 287296, 288369,
        289444, 290521, 291600, 292681, 293764, 294849, 295936, 297025, 298116, 299209, 300304,
        301401, 302500, 303601, 304704, 305809, 306916, 308025, 309136, 310249, 311364, 312481,
        313600, 314721, 315844, 316969, 318096, 319225, 320356, 321489, 322624, 323761, 324900,
        326041, 327184, 328329, 329476, 330625, 331776, 332929, 334084, 335241, 336400, 337561,
        338724, 339889, 341056, 342225, 343396, 344569, 345744, 346921, 348100, 349281, 350464,
        351649, 352836, 354025, 355216, 356409, 357604, 358801, 360000, 361201, 362404, 363609,
        364816, 366025, 367236, 368449, 369664, 370881, 372100, 373321, 374544, 375769, 376996,
        378225, 379456, 380689, 381924, 383161, 384400, 385641, 386884, 388129, 389376, 390625,
        391876, 393129, 394384, 395641, 396900, 398161, 399424, 400689, 401956, 403225, 404496,
        405769, 407044, 408321, 409600, 410881, 412164, 413449, 414736, 416025, 417316, 418609,
        419904, 421201, 422500, 423801, 425104, 426409, 427716, 429025, 430336, 431649, 432964,
        434281, 435600, 436921, 438244, 439569, 440896, 442225, 443556, 444889, 446224, 447561,
        448900, 450241, 451584, 452929, 454276, 455625, 456976, 458329, 459684, 461041, 462400,
        463761, 465124, 466489, 467856, 469225, 470596, 471969, 473344, 474721, 476100, 477481,
        478864, 480249, 481636, 483025, 484416, 485809, 487204, 488601, 490000, 491401, 492804,
        494209, 495616, 497025, 498436, 499849, 501264, 502681, 504100, 505521, 506944, 508369,
        509796, 511225, 512656, 514089, 515524, 516961, 518400, 519841, 521284, 522729, 524176,
        525625, 527076, 528529, 529984, 531441, 532900, 534361, 535824, 537289, 538756, 540225,
        541696, 543169, 544644, 546121, 547600, 549081, 550564, 552049, 553536, 555025, 556516,
        558009, 559504, 561001, 562500, 564001, 565504, 567009, 568516, 570025, 571536, 573049,
        574564, 576081, 577600, 579121, 580644, 582169, 583696, 585225, 586756, 588289, 589824,
        591361, 592900, 594441, 595984, 597529, 599076, 600625, 602176, 603729, 605284, 606841,
        608400, 609961, 611524, 613089, 614656, 616225, 617796, 619369, 620944, 622521, 624100,
        625681, 627264, 628849, 630436, 632025, 633616, 635209, 636804, 638401, 640000, 641601,
        643204, 644809, 646416, 648025, 649636, 651249, 652864, 654481, 656100, 657721, 659344,
        660969, 662596, 664225, 665856, 667489, 669124, 670761, 672400, 674041, 675684, 677329,
        678976, 680625, 682276, 683929, 685584, 687241, 688900, 690561, 692224, 693889, 695556,
        697225, 698896, 700569, 702244, 703921, 705600, 707281, 708964, 710649, 712336, 714025,
        715716, 717409, 719104, 720801, 722500, 724201, 725904, 727609, 729316, 731025, 732736,
        734449, 736164, 737881, 739600, 741321, 743044, 744769, 746496, 748225, 749956, 751689,
        753424, 755161, 756900, 758641, 760384, 762129, 763876, 765625, 767376, 769129, 770884,
        772641, 774400, 776161, 777924, 779689, 781456, 783225, 784996, 786769, 788544, 790321,
        792100, 793881, 795664, 797449, 799236, 801025, 802816, 804609, 806404, 808201, 810000,
        811801, 813604, 815409, 817216, 819025, 820836, 822649, 824464, 826281, 828100, 829921,
        831744, 833569, 835396, 837225, 839056, 840889, 842724, 844561, 846400, 848241, 850084,
        851929, 853776, 855625, 857476, 859329, 861184, 863041, 864900, 866761, 868624, 870489,
        872356, 874225, 876096, 877969, 879844, 881721, 883600, 885481, 887364, 889249, 891136,
        893025, 894916, 896809, 898704, 900601, 902500, 904401, 906304, 908209, 910116, 912025,
        913936, 915849, 917764, 919681, 921600, 923521, 925444, 927369, 929296, 931225, 933156,
        935089, 937024, 938961, 940900, 942841, 944784, 946729, 948676, 950625, 952576, 954529,
        956484, 958441, 960400, 962361, 964324, 966289, 968256, 970225, 972196, 974169, 976144,
        978121, 980100, 982081, 984064, 986049, 988036, 990025, 992016, 994009, 996004, 998001,
        1000000
    ]
    .iter()
    .cloned()
    .collect();
}

#[fastout]
fn main() {
    input! {
        n: usize,
        d: usize,
        x: [[i32; d]; n],
    }

    let mut ans = 0;

    for i in 0..n {
        for j in 0..i {
            let mut distance = 0;
            for d in 0..d {
                distance += (x[i][d] - x[j][d]).pow(2);
            }
            if SQUARES.contains(&distance) {
                ans += 1;
            }
        }
    }

    println!("{}", ans);
}