Leaderboard
New Jersey
Rank
Level
1
DoMo
Somerville
93
2
User
Princeton
93
3
User
Princeton
91
4
Anthony McWilliams
Toms River
91
5
Kyle Xiao
Newark
90
6
User
East Brunswick
86
7
User
Jersey City
86
8
balsu
Mahwah
83
9
User
Paramus
81
10
User
Princeton
81
11
III
Edison
79
12
User
Jersey City
74
13
Anitya Gupta
Harrison
72
14
William Qiu
Jersey City
71
15
User
Voorhees Township
71
16
User
Franklin Township
70
17
User
South Brunswick Township
69
18
User
Hazlet
68
19
User
Newark
66
20
User
South Brunswick Township
62
21
User
Voorhees Township
61
22
User
Hoboken
61
23
User
Harrison
61
24
User
Voorhees Township
59
25
User
Woodstown
59
26
Prachet Trivedi
Matawan
58
27
User
Jackson
57
28
User
Voorhees Township
57
29
User
Newark
57
30
Michael B
Middletown
57
31
Junjie Lin
Englewood
56
32
User
Jersey City
55
33
User
Jersey City
54
34
User
Montgomery
53
35
Mohammed Namazi
Old Bridge Township
52
36
User
Jersey City
52
37
User
Franklin Township
50
38
Christopher Ye
Princeton
50
39
User
Montgomery
48
40
User
Old Bridge Township
48
41
User
Mount Holly
48
42
User
Fort Lee
48
43
Epic Bhangra
Old Bridge Township
46
44
Dinofloss
Brick
46
45
G
West Windsor Township
46
46
User
Hoboken
46
47
User
Harrison
46
48
Ru Poissen
Fort Lee
45
49
User
Newark
45
50
User
Montgomery
44
51
User
Tewksbury
43
52
User
North Brunswick Township
43
53
User
Fort Lee
43
54
Daniel Kim
Hackensack
42
55
User
Jersey City
42
56
User
Englewood
42
57
Audrey
Mount Laurel
41
58
User
Old Bridge Township
41
59
User
Jersey City
41
60
User
Old Bridge Township
40
61
User
Franklin Township
40
62
User
Cherry Hill
39
63
User
Roseland
39
64
Jesse Pierson
Edison
39
65
User
Upper Saddle River
38
66
User
West New York
38
67
User
Franklin Township
38
68
User
Edison
38
69
User
Galloway
38
70
User
Somerset
38
71
User
Princeton
37
72
User
Brick
37
73
User
Manchester Township
37
74
User
Englewood
36
75
Eli Bosnick
Belleville
36
76
User
Jersey City
36
77
User
Princeton
35
78
User
Marlboro Township
35
79
User
East Brunswick
35
80
User
Robbinsville
35
81
Todd Solecitto
Washington Township
35
82
User
West Orange
34
83
Amogh Sarangdhar
Montville
33
84
Justin Frazee
Lacey Township
31
85
User
Old Bridge Township
31
86
User
Wall Township
31
87
Bob Sanderia
Galloway
31
88
User
New Brunswick
31
89
Cindy Hu
West Orange
31
90
User
Highland Park
31
91
User
Bayonne
31
92
User
Princeton
31
93
User
Jersey City
31
94
Divyendra Patil
Newark
31
95
User
Jersey City
30
96
User
Montclair
30
97
User
Monroe Township
30
98
User
Old Bridge Township
30
99
User
Jersey City
30
100
User
Hanover
30
101
User
Linden
30
102
User
Princeton
30
103
User
Old Bridge Township
29
104
User
Madison
29
105
User
Ridgewood
28
106
User
Jersey City
28
107
Joseph Bond MacDonald
Washington Township
28
108
User
Old Bridge Township
28
109
User
Princeton
28
110
User
Cliffside Park
28
111
User
West Orange
28
112
User
Old Bridge Township
28
113
Rahul K
Robbinsville
28
114
User
Secaucus
28
115
User
Hoboken
28
116
User
Trenton
28
117
User
Nutley
28
118
User
Edison
27
119
Mahesh Kshirsagar
West Windsor Township
27
120
Will Spadafora
Montclair
27
121
User
Edison
27
122
User
West Windsor Township
27
123
User
Hopewell Township
26
124
User
Marlboro Township
26
125
User
Hackensack
26
126
User
Cherry Hill
26
127
Matthew Grothusen
Hoboken
26
128
User
Morris Township
26
129
User
Old Bridge Township
26
130
User
Old Bridge Township
26
131
User
Harrison
26
132
User
Princeton
26
133
User
South Plainfield
26
134
User
Jersey City
26
135
User
Hoboken
26
136
User
East Hanover
26
137
User
South Plainfield
26
138
User
Teaneck
25
139
User
Bloomfield
25
140
User
Old Bridge Township
25
141
User
Princeton
25
142
User
Franklin Township
24
143
Cristian Torres
Newark
24
144
User
Newark
24
145
User
Fair Lawn
24
146
User
Elmwood Park
24
147
User
Piscataway Township
24
148
appllo
Tenafly
24
149
Sanket Prasad
Edison
24
150
User
Edison
24
151
User
Princeton
24
152
User
Bellmawr
23
153
User
Newark
23
154
User
Newark
23
155
User
Marlboro Township
23
156
User
Toms River
23
157
User
Cherry Hill
23
158
User
Camden
23
159
User
Morris Plains
23
160
User
Highland Park
23
161
User
Tenafly
23
162
User
Lawrence Township
23
163
User
Belleville
23
164
User
Collingswood
23
165
User
Franklin Township
22
166
Sohum Kshirsagar
West Windsor Township
22
167
User
Millville
22
168
User
Toms River
22
169
User
Clifton
22
170
User
Woodbridge Township
22
171
Myfitness Journey
Montgomery
22
172
User
Mount Laurel
22
173
User
Hammonton
22
174
User
New Milford
21
175
Jose Hernandez-Morel
Newark
21
176
Skyly Ramirez
Newark
21
177
Sarah Cheng
Princeton
21
178
User
Parsippany-Troy Hills
21
179
User
Fair Lawn
21
180
User
Old Bridge Township
21
181
Neelay
Harrison
21
182
User
Princeton
21
183
User
Livingston
21
184
Vishak Chidam
Jersey City
21
185
Vishy C
Jersey City
21
186
User
Pennsauken Township
21
187
User
Mount Laurel
21
188
Cristopher Flores
Newark
20
189
Michael Sorkin
Maplewood
20
190
User
Lodi
20
191
User
Paramus
20
192
Nick H
Princeton
20
193
User
South Brunswick Township
20
194
User
Ridgewood
20
195
User
North Bergen
20
196
User
Newark
20
197
User
Old Bridge Township
20
198
User
Montgomery
19
199
User
Newark
19
200
User
East Hanover
19
201
Elian Polanco
Jersey City
19
202
User
Newark
19
203
User
Princeton
19
204
User
Hanover
19
205
User
Princeton
19
206
User
Clifton
19
207
User
Princeton
19
208
User
North Bergen
19
209
User
East Windsor
19
210
User
Ridgewood
19
211
User
Clifton
19
212
User
Monroe Township
19
213
User
Mahwah
19
214
User
Franklin Township
18
215
User
Paterson
18
216
User
Newark
18
217
User
Ocean City
18
218
User
Hackensack
18
219
User
Bloomfield
18
220
User
Camden
18
221
User
Highland Park
18
222
User
Edgewater
18
223
User
Jersey City
18
224
User
Paramus
18
225
User
Newark
17
226
User
Parsippany-Troy Hills
17
227
User
Howell
17
228
User
Passaic
17
229
User
Bridgewater
17
230
User
Secaucus
17
231
User
Ridgewood
17
232
User
North Bergen
17
233
User
Jersey City
17
234
User
Jersey City
17
235
User
Montgomery
16
236
User
Franklin Township
16
237
Mauricio Marin
Newark
16
238
Sarah Velez
Newark
16
239
User
Newark
16
240
User
Newark
16
241
User
Parsippany-Troy Hills
16
242
User
Milford
16
243
User
South Brunswick Township
16
244
User
Jersey City
16
245
User
Jersey City
16
246
User
North Bergen
16
247
User
Jersey City
16
248
User
Collingswood
16
249
User
Jersey City
16
250
User
Cherry Hill
16
251
User
Beachwood
16
252
User
Vineland
16
253
User
Montgomery
15
254
User
Newark
15
255
User
Randolph
15
256
Sandy Bonilla
Newark
15
257
User
Princeton
15
258
User
Parsippany-Troy Hills
15
259
User
Raritan
15
260
User
Ridgefield
15
261
User
Monroe Township
15
262
User
Cranford
15
263
User
Elmwood Park
15
264
Katherine Laurens
Jersey City
15
265
User
Princeton
15
266
User
Harrison
15
267
User
Vineland
15
268
User
Trenton
15
269
ACE FACE
Edison
15
270
pablo chavez
Hoboken
14
271
User
Franklin Township
14
272
User
Newark
14
273
User
Newark
14
274
User
Franklin Township
14
275
User
Newark
14
276
Daniel Mata Murillo
Newark
14
277
User
Toms River
14
278
User
West Windsor Township
14
279
User
Newark
14
280
User
Ridgefield Park
14
281
User
Hackensack
14
282
User
Linden
14
283
User
Fort Lee
14
284
User
Madison
14
285
User
Highland Park
14
286
User
Piscataway Township
14
287
User
Newark
14
288
User
Princeton
14
289
User
Tabernacle
14
290
User
New Brunswick
14
291
User
Piscataway Township
14
292
User
Monroe Township
14
293
User
Westfield
13
294
User
Newark
13
295
User
Newark
13
296
User
South Brunswick Township
13
297
User
East Brunswick
13
298
User
Passaic
13
299
User
South Brunswick Township
13
300
User
Perth Amboy
13
301
User
Paterson
13
302
Kaira Vazquez
Newark
13
303
User
Newark
13
304
User
Newark
13
305
User
Princeton
13
306
User
Woodbridge Township
13
307
User
Princeton Meadows
13
308
User
Flemington
13
309
User
Jersey City
13
310
User
Maplewood
13
311
User
Summit
13
312
User
New Hanover Township
13
313
User
Highland Park
13
314
User
East Brunswick
13
315
User
Jersey City
13
316
User
Brick
13
317
User
New Providence
13
318
User
Voorhees Township
13
319
User
Newark
12
320
User
East Brunswick
12
321
User
Edison
12
322
User
North Bergen
12
323
User
Sparta Township
12
324
User
Paterson
12
325
User
Toms River
12
326
User
Fort Lee
12
327
User
Newark
12
328
User
Newark
12
329
User
Newark
12
330
Lewis Lantigua
Newark
12
331
User
Asbury Park
12
332
A J
Newark
12
333
User
Jersey City
12
334
User
Edison
12
335
User
Princeton
12
336
User
Newark
12
337
User
Highland Park
12
338
User
Piscataway Township
12
339
User
Edison
12
340
User
Mount Laurel
12
341
User
Kearny
12
342
User
Elizabeth
12
343
User
Ridgewood
12
344
User
East Brunswick
11
345
User
Ridgewood
11
346
User
Franklin Township
11
347
User
Harrison
11
348
User
Bridgewater
11
349
User
Newark
11
350
User
Elizabeth
11
351
User
Livingston
11
352
User
Jersey City
11
353
User
Weehawken
11
354
Erick Romano
Newark
11
355
User
Newark
11
356
Oscar Recinos-Giron
Newark
11
357
Teoeuroi
Jersey City
11
358
User
Oakland
11
359
User
Bloomfield
11
360
User
Cliffside Park
11
361
User
Jackson
11
362
User
City
11
363
User
Newark
11
364
Subharthi Sen
Princeton
11
365
User
Princeton
11
366
User
South Plainfield
11
367
User
Newark
10
368
User
Hazlet
10
369
User
Edison
10
370
CIRA
Newark
10
371
User
Newark
10
372
User
Asbury Park
10
373
andress etindele
Secaucus
10
374
User
Ridgefield Park
10
375
User
Newark
10
376
User
Newark
10
377
User
Newark
10
378
User
Westfield
10
379
User
Toms River
10
380
User
Jersey City
10
381
User
Clifton
10
382
User
North Caldwell
10
383
User
Marlboro Township
10
384
User
Mahwah
10
385
User
Highland Park
10
386
User
Highland Park
10
387
User
Highland Park
10
388
User
Highland Park
10
389
User
Tinton Falls
10
390
User
River Edge
10
391
User
Princeton
10
392
User
Westwood
10
393
User
Middletown
10
394
Nyce
Trenton
10
395
User
Wayne
10
396
User
City
9
397
User
Cherry Hill
9
398
User
Raritan
9
399
User
Montgomery
9
400
User
Franklin Township
9
401
Jack F
Newark
9
402
Isiah Medina
Newark
9
403
User
Newark
9
404
User
East Brunswick
9
405
User
Newark
9
406
User
Newark
9
407
User
Newark
9
408
User
West Orange
9
409
Carol Lorac
City
9
410
User
Oakland
9
411
User
Paterson
9
412
User
Edison
9
413
User
Newark
9
414
Victor Jhoan Bonilla
Newark
9
415
User
Newark
9
416
User
Newark
9
417
User
Freehold Township
9
418
User
West Long Branch
9
419
User
West Windsor Township
9
420
User
Monroe Township
9
421
User
Woodbridge Township
9
422
User
Jersey City
9
423
User
Bergenfield
9
424
User
Lopatcong
9
425
User
Ho-Ho-Kus
9
426
User
Union
9
427
User
Jersey City
9
428
User
Cresskill
9
429
User
Livingston
9
430
User
Monroe Township
9
431
User
Bayonne
9
432
User
Morristown
9
433
User
Jackson
9
434
User
City
8
435
Marbely Martinez
Newark
8
436
User
Raritan
8
437
User
Newark
8
438
User
Newark
8
439
Josef Kariuki
Union
8
440
User
Newark
8
441
User
Newark
8
442
User
Edison
8
443
User
Montvale
8
444
User
Sparta Township
8
445
User
Newark
8
446
Cira Robinson
Newark
8
447
polly
Bridgewater
8
448
User
Newark
8
449
User
Egg Harbor Township
8
450
User
Brick
8
451
User
Old Bridge Township
8
452
User
Jersey City
8
453
User
Ridgefield Park
8
454
User
Newark
8
455
Camilo Aguilar
Newark
8
456
Sai Villegas
Newark
8
457
User
North Bergen
8
458
User
Newark
8
459
User
Newark
8
460
User
Newark
8
461
User
Flemington
8
462
User
Monroe Township
8
463
Julinel Rodriguez
Newark
8
464
Carlos Gomez
Newark
8
465
User
Newark
8
466
User
Newark
8
467
Camilo Aguilar Espitia
Newark
8
468
User
Newark
8
469
User
Newark
8
470
User
Newark
8
471
User
Jersey City
8
472
User
Bergenfield
8
473
User
Glassboro
8
474
User
Holmdel
8
475
User
Hawthorne
8
476
User
Newark
8
477
User
Parsippany-Troy Hills
8
478
User
Highland Park
8
479
User
Irvington
8
480
User
Princeton
8
481
User
Ridgewood
8
482
User
Voorhees Township
8
483
User
Monroe Township
8
484
User
Livingston
7
485
Debbus Q
Evesham Township
7
486
User
Holmdel
7
487
User
Holmdel
7
488
User
Franklin Township
7
489
User
Montgomery
7
490
User
Newark
7
491
User
Woolwich Township
7
492
User
Pennsauken Township
7
493
User
Newark
7
494
User
Princeton
7
495
User
Irvington
7
496
User
Clifton
7
497
User
Hamilton Township
7
498
User
Manalapan Township
7
499
User
North Caldwell
7
500
User
Paterson
7
501
User
North Brunswick Township
7
502
User
Randolph
7
503
User
Piscataway Township
7
504
User
Jersey City
7
505
Sal Cohen
Egg Harbor Township
7
506
Wendy Cedillos
Newark
7
507
User
Newark
7
508
User
Newark
7
509
User
Jersey City
7
510
User
Newark
7
511
User
Toms River
7
512
Katia Nieves
Newark
7
513
User
Newark
7
514
User
Newark
7
515
User
Newark
7
516
User
Newark
7
517
User
Jackson
7
518
User
Jersey City
7
519
User
Edison
7
520
User
Fair Lawn
7
521
User
Westfield
7
522
User
Summit
7
523
User
Highland Park
7
524
User
Highland Park
7
525
User
Galloway
7
526
User
Newark
7
527
User
Newark
7
528
User
Burlington Township
7
529
User
Bellmawr
7
530
User
Lambertville
7
531
User
Sayreville
7
532
User
Bridgewater
7
533
User
Edison
7
534
User
Nutley
7
535
User
Vineland
7
536
Michael Yang
Jersey City
6
537
User
Newark
6
538
User
Franklin
6
539
User
Newark
6
540
User
Montgomery
6
541
Pac-Man
Holmdel
6
542
User
Holmdel
6
543
User
Newark
6
544
Ismail Richardson
Newark
6
545
User
Newark
6
546
User
Newark
6
547
User
Newark
6
548
User
South Plainfield
6
549
User
Newark
6
550
User
Newark
6
551
User
Morristown
6
552
User
Monroe Township
6
553
User
Bridgewater
6
554
User
Newark
6
555
User
Hoboken
6
556
User
Paterson
6
557
Keycy Montano🇸🇻🎮⚽️
Newark
6
558
ALESSANDRO CERVERA
Paterson
6
559
User
Franklin Township
6
560
Angelin Castro
Newark
6
561
User
Newark
6
562
User
Newark
6
563
User
Newark
6
564
User
Newark
6
565
User
Newark
6
566
User
Newark
6
567
User
Newark
6
568
User
Newark
6
569
User
Newark
6
570
User
Clifton
6
571
User
North Bergen
6
572
User
West Long Branch
6
573
User
New Brunswick
6
574
User
Paterson
6
575
User
Montgomery
6
576
User
Woodbridge Township
6
577
Sesilja ?
Plumsted Township
6
578
User
Newark
6
579
User
Mount Laurel
6
580
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6
581
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6
582
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6
583
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6
584
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6
585
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586
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6
587
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6
588
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5
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5
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5
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5
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5
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5
603
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5
605
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5
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5
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5
608
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5
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5
610
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5
611
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5
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5
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5
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5
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5
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5
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5
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5
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5
620
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Newark
5
621
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5
622
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Newark
5
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5
624
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5
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Newark
5
626
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Newark
5
627
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5
628
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Newark
5
629
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Newark
5
630
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Newark
5
631
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Newark
5
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Newark
5
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Newark
5
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Newark
5
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Newark
5
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5
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5
638
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Middletown
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639
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5
640
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641
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642
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643
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5
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645
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5
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647
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5
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5
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Hightstown
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4
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4
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4
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4
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4
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4
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4
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4
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4
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4
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4
694
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Newark
4
695
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4
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4
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Newark
4
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Newark
4
699
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Newark
4
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Newark
4
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Newark
4
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Newark
4
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Newark
4
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Newark
4
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Newark
4
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Newark
4
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4
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709
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4
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4
711
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4
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Newark
4
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Newark
4
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Newark
4
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Newark
4
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4
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4
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4
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4
720
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4
721
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4
722
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4
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4
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4
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4
726
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4
727
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4
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4
729
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4
730
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4
731
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4
732
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733
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734
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4
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4
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4
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4
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4
741
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4
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4
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4
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4
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3
746
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Newark
3
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Newark
3
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Newark
3
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Newark
3
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3
751
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3
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3
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3
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3
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3
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3
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3
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3
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3
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3
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3
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3
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3
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Paterson
3
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Newark
3
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Newark
3
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Newark
3
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Newark
3
771
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Newark
3
772
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Newark
3
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Newark
3
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Newark
3
775
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Linden
3
776
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Newark
3
777
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3
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3
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3
780
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3
781
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3
782
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3
783
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3
784
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Newark
3
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Newark
3
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Newark
3
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Newark
3
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Newark
3
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Newark
3
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3
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3
792
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3
793
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3
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Newark
3
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Newark
3
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Newark
3
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3
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3
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3
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3
801
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3
802
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3
803
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3
804
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3
805
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3
806
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3
807
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3
808
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3
809
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3
810
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3
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812
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3
813
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3
814
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3
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3
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3
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3
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3
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3
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3
821
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3
822
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3
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3
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3
825
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3
826
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3
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3
828
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3
829
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3
830
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3
831
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3
832
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3
833
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3
834
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Jersey City
3
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3
836
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3
837
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3
838
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3
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3
840
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3
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3
843
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3
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3
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3
846
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3
847
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3
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3
849
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3
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3
851
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3
852
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2
853
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2
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2
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2
862
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2
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2
866
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2
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2
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2
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2
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Newark
2
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Newark
2
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2
873
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2
874
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2
875
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2
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2
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2
878
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879
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2
880
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2
881
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2
882
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2
883
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2
884
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2
885
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2
886
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2
887
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2
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2
889
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2
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2
891
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2
892
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2
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2
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2
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2
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2
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2
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2
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2
900
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2
901
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2
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2
903
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2
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Newark
2
907
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2
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Newark
2
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2
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2
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2
913
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2
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2
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Newark
2
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2
918
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Newark
2
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Newark
2
920
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Newark
2
921
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Newark
2
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Newark
2
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Roselle
2
924
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Newark
2
925
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2
926
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2
927
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2
928
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2
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2
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2
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2
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2
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2
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2
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2
936
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2
937
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2
938
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2
939
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2
940
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2
941
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2
942
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2
943
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2
944
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2
945
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2
946
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2
947
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2
948
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2
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2
950
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2
951
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2
952
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2
953
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2
954
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2
955
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2
956
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2
957
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2
958
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2
959
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2
960
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2
961
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2
962
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2
963
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2
964
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2
965
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2
966
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2
967
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2
968
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2
969
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2
970
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2
971
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2
972
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2
973
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2
974
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2
975
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2
976
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2
977
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2
978
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2
979
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2
980
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2
981
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2
982
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2
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2
984
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2
985
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2
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2
987
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2
988
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2
989
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2
990
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2
991
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992
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2
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2
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997
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998
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2
999
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A Web App for Mathematics Training

Do you want to be fast at mental math? Many people do, but the options for doing the necessary exercises are simply too cumbersome for all but the most dedicated of trainees. In physical fitness, many people are interested in training their bodies but allocating the time, energy, and money for it is a significant obstacle. Likewise, lugging around books and whatnot for math practice is a threshold that just doesn't meet the standards of modern life.

Training yourself to be skilled at mental math needs to be quick and convenient. mathtrainer.org is a web app that works in your browser rather than a program you have to download and install on your computer or phone. This allows users to try and use the app without having to install new software. As a web app, updates are also much simpler. There is no need to download endless updates—the website will always be the most current version.

You can access a web app from any device connected to the internet and equipped with a web browser, including smartphones, tablets, and desktop computers. Moreover, you are free to use whichever browser you prefer, including Google Chrome, Safari, Firefox, and others. Google Chrome is the recommended browser for the best maths training since it tends to lead the pack in supporting the latest web technologies.

Math Trainer is designed to offer a similar experience regardless of what you’re using to access it, whether it be Android, iOS, Windows, or another operating system. Though an on-screen touch keyboard will appear on mobile devices, you may prefer to use the app on a desktop with a keyboard. Hopefully the advantages of a web app for convenient mathematics training are apparent.

Another part of making the app easy to use is eliminating the need for signing up and logging in. Users can get started with their math training as soon as they click the start button on this page. After progressing to higher levels in the app, your progress is automatically saved so long as your return to the site through the same browser.


What's a Mental Math Tip?

A mental math tip is a sequence of steps that can be taken to solve a math problem in your head. Click the arrow below to see an example for the following problem:

÷
984
3
328

A tip like this one is available for every problem in Math Trainer, so there's always help if you get stuck. With enough practice, you'll be able to predict what the tips will say—you'll have learned mental math!


Get Better at Mental Math

The ability to quickly perform mental calculations offers advantages in certain circumstances. But even without applications, getting better at mental math is a great way to stimulate one’s mind. It develops better number sense and intuition for quantifying the world around us. Practicing mental calculation will strengthen your foundation for learning more advanced maths topics.

Nonetheless, the tangible benefits of improving at mental math are many. It is certainly expected that educated people are able to do simple arithmetic without having to pull out a calculator. An inability to do so may reflect poorly on you, while being well-practiced in mental calculation will leave your contemporaries impressed. In many scientific and technical circles, mental math ability is even more highly regarded.

For students, mental calculation speed will often have a direct impact on math and science test scores. At all grade levels, it is not sufficient to know how to solve math problems when tests have a time limit on them. The highest-scoring test takers are able to answer questions both correctly and efficiently. Improving mental math skills will only benefit a student’s academic career.

Calculating the solution to an arithmetic problem in your head is often faster than pulling out a device to tell you the answer. For example, figuring out how much to tip a server at a restaurant is a straightforward arithmetic problem that many people are unable to perform without a calculator. By training your brain to solve basic math problems, you can save time in situations like these.

Mental math can also be relied upon when calculation devices are not available. Even with the conveniences of modern life, we occasionally find ourselves without access to our cell phones or other capable devices. A mind skilled in mental math is always available to you.

Finally, getting better at mental math enables a quick estimate and sanity check on results obtained from calculators. While computers are extremely reliable at solving math problems, there is always the risk of incorrectly inputting the problem to the computer. By getting better at mental mathematics, you will develop an intuition for whether the results of calculators make sense.

In fact, the ability to estimate is often sufficient to avoid using calculators altogether. While the use of computers is widespread, estimation is an increasingly valued skill in many industries. There are many situations where complex math will eventually be required, but a preliminary estimate is needed quickly. A major boost to productivity!


Use a Math Trainer

Mental math ability is a lot like physical fitness training. You may be out of shape in the beginning, but with diligent training you can and will improve. Initially you might not enjoy the exercise, but you will reap significant rewards for your effort. As you become more fit, you’ll begin to enjoy the activity much more. If you are serious about it, your mental calculation fitness could become a source of energy, galvanizing you to face the challenges of life with enthusiasm.

In physical training, you break down the fibers in your muscles during a workout session. Your muscles actually sustain tiny tears during resistance training exercises. While you rest afterwards, your body repairs the damage, rebuilding the fibers thicker and stronger.

A similar process is believed to occur for cognitive tasks. A 2016 study found "extensive evidence that brain-training interventions improve performance on the trained tasks".1 Therefore you can expect training your brain to answer mental math questions will lead to improved performance over time.

In the context of physical fitness, a "trainer" often refers to a trained professional who guides the workout and recovery process. Personal trainers are tasked with assessing a trainee's level of ability, prescribing an exercise regimen, and offering feedback as the training goes along. The word "trainer" could also refer to a system that automates the role of a personal trainer. Many aerobic exercise machines today offer interactive training programs with feedback and analysis of a user's performance.

A math trainer is needed for optimal math fitness. Like in physical fitness, the trainer should be compatible with users at a variety of skill levels and should guide them to the next level. It should give an accurate assessment of a user's strengths and weakness, as well as offer helpful feedback on where to focus one's efforts. Learning the ropes of mental maths with a math trainer should be a seamless, rewarding journey to ever-greater abilities.


Mental Calculation

Mental calculation, or mental math, is performing arithmetical calculations without the aid of tools or supplies. As opposed to using a calculator or pencil and paper, mental math is performed entirely in one’s head.

People use mental calculation when computation aides are not available, when it is faster to do so, or when they wish to practice, show off, or participate in mental math competitions. Most people perform basic mental calculation using elementary arithmetic on a daily basis. An inability to calculate mentally is a serious obstacle to many common tasks.

In U.S. schools, mental calculation is taught only for the most elementary arithmetic, such as single-digit addition and multiplication of two numbers between 0 and 12. To solve addition problems involving multiple digits, you are taught to add columns of digits from right to left, carrying the tens digit if the column sum exceeds 9. For example, how would you approach this addition problem?

Example of two-digit
addition problem

If you were trained like many of us were, you’d add the right column to obtain 12. Since that’s two digits, you’d write the 2 under the right column and carry the 10 by writing a 1 above the left column. Finally, you’d add the two tens digits and the carried 1 to obtain the answer, 52.

To solve an addition problem mentally, it’s best to add the columns from left to right. In our example, you could add the tens digit of the second number, 30, to the first number, 14, to obtain 44. This is easier than the full problem because you’re just doing one mental calculation and tacking on the 4 from the 14 as the singles digit. Then you’d add the remaining ones digit of the second number, 8, to 44 to arrive at the answer, 52.

Which approach seems simpler to you? Can you do the first approach without pulling out a pencil and paper? It turns out the same advantages of left-to-right addition apply to much larger numbers as well. It’s unlikely that difficult addition problems can be solved right to left without needing to write it all out, which of course is more time consuming.

Mental math should be distinguished from the memorization of math facts such as multiplication tables. A foundation of memorized answers to simple math problems will make mental math easier, but performing maths in your head requires both memorized facts and the manipulation of numbers and operations to solve problems. This combination of skill and memory allows us to solve far more complex math questions than can be answered with readily-memorized math facts.

Many mental math tricks are specific to particular numbers or types of problems, usually dependent on the base of the number system used. In the decimal numeral system, for example, it is trivially easy to multiply by 10—just add a 0 to the end of the number. This mental math trick wouldn’t work in the hexadecimal numeral system, though, because the base is 16 instead of 10.

Therefore mental calculation is the ability to manipulate complex arithmetic problems in such a way that they can be resolved using simple memorized math facts.


Arithmetic

Arithmetic is the branch of mathematics concerning basic number operations: addition, subtraction, multiplication, and division. As kids, we are taught to do arithmetic because real-world math problems depend on a mastery of elementary arithmetic. Higher-level study of arithmetic and the integers, or whole numbers, is known as number theory.

Though the math kids initially study is arithmetic, the word is rarely used in this context anymore. Originally it comes from the Greek arithmos, meaning “number”. It has however been included in the “three Rs” of elementary Western education: reading, writing, and arithmetic.

There is evidence prehistoric humans were using arithmetic as hunter-gatherers. Archaeologists have uncovered a tally stick, believed to be over 20,000 years old, which may exhibit the earliest known sequences of prime numbers. An understanding of prime numbers, which are only divisible by themselves and the number 1, requires knowledge of the operation in arithmetic known as division.

From tally marks came base-10 numerals such as those used in Egypt over 5,000 years ago. Number systems based on 10 probably arose because humans have ten “digits” as fingers on their hands (or toes on their feet). A later advance in arithmetic was positional notation, which allowed the same symbols to represent different magnitudes depending on their position in the written number. These numeral systems allowed complex arithmetic to be communicated, recorded, and applied to the challenges faced by our ancestors.

The basic operation of arithmetic is addition. It combines two or more numbers into one, the sum of the terms. The terms can be added in any order, which is known as the commutative property of arithmetic. On a number line, the sum of two numbers is the total distance from zero covered by both numbers.

The inverse arithmetical operation of addition is subtraction. It finds the difference between two numbers. Subtraction is not commutative because the order of the numbers determines whether the answer is positive or negative. On a number line, the difference between two numbers is the distance between their positions.

A second basic operation of arithmetic is multiplication, which scales a number by another number. This second number is called a factor. Like addition, multiplication is commutative—you can change the order of the factors and still get the same answer. Multiplication on a number line can be viewed as adding the first number a total number of times equal to the second factor.

Finally, division is an arithmetical operation that is essentially the inverse of multiplication. Rather than scaling a number, it is divided into a number of pieces equal to the second number. Dividing by the number 0 is not defined in arithmetic because dividing something into zero pieces is impossible.

Basic arithmetic allows us to evaluate the answers to an unlimited number of mathematical expressions. Arithmetical expressions can be purely mathematical, as in 2 + 2, or they can represent quantities in the physical world, such as two items plus two more. Understanding the laws of arithmetic is tremendously powerful.


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