Number Pattern
Journal of Recreational Mathematics. (2010
Number Pattern
Hui Fang Huang Su 0
0 Nova Southeastern University
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Article 5
Cover Page Footnote
This article is the result of the MAT students' collaborative research work in the PreAlgebra course. The
research was under the direction of their professor, Dr. Hui Fang Su. The paper was organized by Team Leader
Denise Gates.
Authors
Hui Fang Huang Su, Denise Gates, Janice Haramis, Farrah Bell, Claude Manigat, Kristin Hierpe, and
Lourivaldo Da Silva
This article is available in Transformations: http://nsuworks.nova.edu/transformations/vol2/iss2/5
Number Patterns
Abstract
In this manuscript, we study the purpose of number patterns, a brief history of number
patterns, and classroom uses for number patterns and magic squares. We investigate and
summarize number patterns and magic squares in various charts: 6 x 6, 7 x 7, 13 x 13, 21 x 21,
and 37 x 37. The results are established by each number pattern along with narrative conjectures
about primes and multiples of six from each pattern. Numerical charting examples are provided
as an illustration of the theoretical results.
A Brief History of Number Patterns
Recognizing number patterns is a vital problemsolving skill. As noted by the Annenburg
Foundation, “If you see a pattern when you look systematically at specific examples, you can use
that pattern to generalize what you see into a broader solution to a problem”
(Annenburg
Foundation, 2016)
. Understanding number patterns are necessary so that students of all ages can
appropriately identify and understand various types of patterns and functional relationships.
Furthermore, number pattern awareness allows one to use patterns and models to analyze the
change in both real and abstract contexts. The Common Core State Standards state that
“mathematically proficient students look closely and discern a pattern or structure” (CCSS,
2015). In addition to the Common Core State Standards, the National Council of Teachers of
Mathematics states that “In prekindergarten through grade 2 all students should use multiple
models to develop initial understandings of place value and the baseten number system”
(NCTM, 2015).
How can number sense and number pattern awareness be developed and or enhanced
upon? A hundreds chart can be used to provide students with a framework for thinking about the
base ten number system; it allows students to develop a mental model of the number system.
Utilizing a hundreds chart will provide students with the opportunity to look for and make sense
of number patterns and structure within the base ten number system. Furthermore, the familiarity
with a hundreds chart will also build upon a student’s sense of number patterns and awareness,
and, therefore, lead to computational flexibility and fluency. A hundreds chart can be used for a
variety of activities. Using a hundreds chart, students can look for number patterns, they can
brush up on their addition and subtraction facts, they can build their multiplication sense, they
can broaden their knowledge of fractions and decimals and enhance their logical and strategic
thinking skills (Gaskins, 2008).
15 15 15
There are several other ways to array the numbers in each of the nine cells to create a 3x3
magic square. Given the task of arranging consecutive numbers in a pattern so that the rows and
15
columns form an equal sum, one can add the consecutive numbers and divide by the number of
columns or rows. In the example above, 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 / 3 = 15 are the
sum of the rows, columns, and diagonals in the magic square. It can be an exercise to the student
– if the student can determine the sum established on the number of rows and columns in the
square, he or she can arrange consecutive numbers in the square appropriately. The larger the
square, the more cumbersome this methodology becomes, which is why mathematicians have
created the following formula:
In the above formula, n symbolizes the number of rows and columns. If we continue to
use the example of a 3 x 3 square, the formula would simplify to (3(32 + 1) / 2) = 15.
Understandably, the larger n becomes larger than the sum, thus creating possible arrangements
throughout the square (Magic Squares  What are they).
The origin of the magic square can be traced back to 2800 B.C. Chinese Literature and
the legend of “Lo Shu” which can be translated to the scroll of the River Lo. The legend tells the
story of a flood that destroyed the land. The people of the city believed that offering a sacrifice to
the river god would calm his anger and keep him from causing further destruction; however the
river god continued to flood the land. The legend states that a turtle would emerge from the river
after every flood and walk around the sacrifice. After this had happened several times, a child
noticed a unique pattern on the turtle’s shell. This pattern, which
can be seen in the figure to the right, told the people how many
sacrifices to make for the river god to calm the waters – fifteen.
Other accounts of this tale state that it was the great Emperor Yu
who was the one to notice this pattern –whoever it was is unclear, but each account does state
that the fifteen sacrifices pleased the river god, and the flooding was ceased (Magic Squares
What are they).
Early Chinese culture is rich with the usage of magic squares, including astrology,
divination, philosophy, natural phenomena, and human behavior. Magic squares and their uses in
various areas of study traveled from China to other parts of Asia and Europe throughout ancient
civilization. Its history is rich, and the square has journeyed a long period. Some of its greatest
achievements in India’s history include Varahamihira, who used a fourthorder magic square to
specify recipes for making perfumes that allowed him to see into the future; and the doctor
Vrnda, who claimed the magic square helped him develop a means to ease childbirth (Anderson).
The ancient Arabic description for magic squares, wafq ala’dad, means “harmonious
disposition of the numbers.” The idea is exemplified by Camman, who speaks of the spiritual
importance of these magical puzzles:
“If magic squares were, in general, small models of the Universe, now they could be
viewed as symbolic representations of Life in a process of constant flux, constantly being
renewed through contact with a divine source at the center of the cosmos.”
(Prussin 1986,
p. 75)
Much of ancient history reveals continued to awe and reverence for the magic square –
Ancient artifacts from Africa to Asia show that the magic square became interwoven into
cultural artifacts, appearing on antique porcelains and sculptures, even in the design and building
of homes (Anderson).
It continued throughout history until the seventeenth century, when French aristocrat
Antoine de la Loubere began to study the mathematical theory behind the construction of magic
squares. In 1686, Adamas Kochansky created a threedimensional magic square. Today, magic
squares are examined about factor analysis, combinational mathematics, matrices, modular
arithmetic, and geometry (Anderson).
6 x 6 Chart – Number Pattern
During number pattern observation, it is highlighted that number patterns are evident. These are
some indication:
i.
2nd, 4th, and 6th columns are all divisible rule of 2 and multiples of 4, which
indicates the multiple of 4 table.
ii.
iii.
iv.
vi.
The table shows multiple of 6, which displays the last column numbers that are
divisible by 6 and multiples of 6.
The 3rd column numbers are divisible by 3, which is indicated in black.
1st, 3rd, and 5th columns are odd numbers, also the 2nd, 4th, and 6th columns are
even numbers.
Each number arrangement shows the columns are increasing by 6. The table shows numbers are prime numbers on the 6 x 6 chart.
6 x 6 Chart – Number Pattern
During number pattern diagonal observation, looking at highlighted ray lines we can see patterns
arrays:
vii.
Looking at the above chart, the corners of the number arrangements on opposite
ends, we can see a pattern of the sum of 37. For example, 6 + 31 = 37 and 1 + 36
Using orange ray lines in Figure 2, the diagonal will display a formula of n + 5.
This pattern increased by five from top right to bottom left. For example, if we
use n = 5, then the formula n + 5 is 5 + 5 = 10, which makes the next diagonal
number arrangement 10. The pattern is the same for all diagonal numbers from
top right to bottom left. However, if we reverse the indicated orange ray line,
then we can see different formula n – 5 using the integer rule. The diagonal going
upward from bottom left to right shows another pattern arrangement. If n = 13,
then 13 – 5 = 7, which is the next diagonal number arrangement. The orange
arrow indicates the pattern formulas n + 5 and n – 5; in fact, we can use it to
demonstrate integer rules in a pattern observation.
ix. Looking at the above chart, the indicated blue arrow going diagonally will
display a formula of n + 7 and n – 7, which demonstrate integer rules patterns
observation. For example, if we use n = 36, then the formula 36 – 7 decrease by
seven from going upward from bottom right to top left. The sum of the next
diagonal number from top left to bottom right is 29. Also, if n = 13, then 13 + 7 =
20, which is the next diagonal number arrangement increased by 7. The pattern is
the same for all diagonal numbers from top left to bottom right.
6 x 6 Chart – Magic Square
During number pattern observation, looking at the magic square we can see patterns arrays:
All rows, columns, and diagonals must add up to the magic square constant of
111 for a 6x6 board. The formula is ! + 1 /2.
The solution is ! !!!!! , ! !!" , !!!! , 111.
7 x 7 Number Patterns
7 x 7 Chart  Observations of the Number Patterns: i. ii.
The values in the 7th (and last) column are all multiples of 7 and divisible by 7.
The columns headed by an odd number alternate odd and even numbers until it reaches
the bottom; the columns headed by an even number are exactly the opposite, alternating
even and odd numbers until the bottom listed number.
iii.
iv.
v.
vi.
In each column, the numbers are increasing by seven from top to bottom.
The table shows prime numbers in the 7x7.
The diagonal going from bottom left to the upper right is n6, upper right to lower left
n+6, The diagonal going from lower right to upper left is n8, upper left to lower right is
n+8.
There is a congruency in the sum of the numbers at opposite ends of the grid:1+49=50
and 7+43=50.
7 x 7 Chart – Observation of the Magic Square:
vii.
viii.
For a 7x7 grid, all rows, diagonals, and columns must add up to the magic number of
175; formula ! + 1 /2. The solution is ! !!!!! , ! !!" , !"!# , 175.
As noted with highlighting, every 7 numbers follows a diagonal pattern.
13 x 13 Chart  Number Patterns
The number 13 is unique in many ways, aside from being a very significant number and
the seventh odd number. The number 13 it is also a part of one of the Pythagorean triples (13,
84, 85) (2010). Below is a 13 x 13 chart I constructed, starting with number 1. Given this
information of the quantity and structure (using numbers 1 through 169, since 13x13=169), the
conjecture would be that there is always, at least, one prime per row, and there is always, at least,
two multiples of six per column.
Moreover, I have identified the prime numbers (pink) and multiples of six (blue). From
the 13x13 chart, one can recognize that there are a few patterns throughout. For instance, there is
a diagonal pattern with the numbers 66, 54, 42, 30, 18, and 6 since they are divisible by six and
increase in odd increments when divided: 11, 9, 7, 5, 3, and 1. A similar pattern takes place in
the first column row 12; beginning with 144, then 132 all the way to 24 and 12. This diagonal
pattern also has multiples of six, but they are in increments of 12 (144/6=24, 132/6=22,…
24/6=4, and 12/6=2) and decreases by two every time. There is an additional correlation with
the numbers 1, 13, and 169 in three corners of the chart as skewrelated cells. Furthermore, two
Pythagorean triangles were identified in the first row, using numbers: 3, 4, 5 and 5, 12, 13.
In 1963, Simon de la Loub’ere (16421729), a mathematician, recognized an algorithm to
construct an odd order square. The pattern began with 1 in the central lower cell, and then
continues diagonally upward to the right in the next column. The next digit, 3, is placed
diagonally downward to the right of 2, and this continues for 3, 4, 5, 6, and 7. The remarkable
chart below for the 13x13 pattern follows this format and can lucidly see the patterns of sequence
organized in an arrangement of colors (Danesi).
Another mastermind puzzlist, who was a prison inmate at the time, created a 13x13
magic square with 11x11 and 3x3 nested inside. The Journal of Recreational Mathematics
published this piece noting that each square is exactly 10,874 smaller than the last, and every cell
is prime (Journal, 2010).
Over time, numerous varieties of patterns, including magic squares, were created as a
spiritual power. These influences stem from Hermetic geometry, where the illustrations
symbolize extraterrestrial shapes. Becoming familiar with these unique shapes and patterns,
such as geometric shapes and the Pythagorean Theorem, can help lay a foundation that students
will utilize in future studies.
21 x 21 Chart  Number Patterns
The number twentyone has several significant accolades to its name. It is a Fibonacci
number and a Harshad number, which is an integer that is divisible by the sum of its digits when
written in that base. It is also the sum of the first six natural numbers, earning the title of a
triangular number. Additionally, twentyone is and an octagonal number, a composite number,
with its divisors being 1, 3, and 7 (all prime), and a Motzkin number (Numbermatics, 2016).
Below is a 21 x 21 chart, starting with number 1. Given this information of the quantity
and structure (using numbers 1 through 441, since 21 x 21 = 441), there are several apparent
patterns throughout this chart each, highlighted in different colors for ease of reading.
21 x 21 Chart – Multiples of 6
In the chart above, multiples of 6 are highlighted in pink to show that in every third row,
every other number is a multiple of 6.
21 x 21 Chart – Prime Numbers
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357
358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399
400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441
In the chart above, prime numbers are highlighted in maroon. Although it 's challenge to
determine a precise pattern for primes, it is interesting to note that at least each column contains,
at least, three prime numbers, and of course, 3 is a factor of 21.
Additional patterns for the 21 x 21 number chart can be found below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357
358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399
400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441
21 x 21 Chart – Multiples of 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357
358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399
400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357
358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399
400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441
21 x 21 Chart – Multiples of 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42
43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147
148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168
169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210
211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231
232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294
295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315
316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336
337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357
358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378
379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399
400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420
421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441
The figure above is a 21 x 21 magic square. Each of the rows, columns, and diagonals
will add to 4,641. A magic square can be found by either adding each of the rows, columns, and
diagonals, or by using the formula (n(n2 + 1) / 2). To create this magic square, the number 1 is
placed in the middle of the upper column (highlighted baby blue), numbers are then “wrapped
around” the square vertically and horizontally. If you look closely, you will see that the numbers
121 are highlighted in baby blue, following a diagonal pattern. Numbers 2242, highlighted in
purple, also follow this pattern. In fact, every 21 numbers (4363, 6484, 85105, etc.) will follow
this exact pattern, creating a truly impressive puzzle.
The pattern of 37 will fill out with 1369 boxes since it’s consists of 37 by 37 or n by n
table. One of the advantages of analyzing number patterns is that often it is possible to see a
small part of it.
There are few easy patterns to find out on the 37 by 37 table. The first one is that the right
diagonal is increased by 36 from 1 until 1369. For instance, (37 + 36 = 73, 73 + 36 = 109, 1297 +
36 = 1333). On the other hand, the left diagonal is augmented by 38. For example, (1 + 38 = 39,
39 + 38 = 77, 1331 + 38 = 1369). Also, it is possible to notice that subtracting any number
starting from the second row from the previous one the result will always be 37. For instance,
(38 – 1 = 37, 704 – 667 = 37, 1060 – 1023 = 37, 962 – 925 = 37, 1369 – 1332 = 37). Also, the
third row is equal to the multiplication of the second row minus the first row. Row3 = R2 x 2 –
R1: (79 = 42 x 2 – 5, 99 = 62 x 2 – 25, 111 = 74 x 2 – 37)
According to the author Chris K.
Caldwell (2015)
, there are plenty conjectures related to
prime number. One of them says that every even n > 2 is the sum of two primes which comes
from the mathematician Goldbach’s work. Back in 1742, Goldback sends a mathematical proof
in a letter to Euler suggesting that every integer n > 5 is equal to the sum of three prime’s
numbers. After Euler analyzing his friend proposal, he found out that every even number greater
than two is equal to the sum of two prime numbers. Also, another important prime conjecture is
more familiar. In fact, there are infinitely many twin primes such as “the sum of the reciprocals
of the twin primes converges, as so the sum B = (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + (1/17
+ 1/19) + is Brun's constant.”
The conjecture formula above proved that there are about twin primes less than or equal
to x, and it also shows that there is an infinite product of the twin primes constants
(Caldwell, C.
K., 2015)
.
multiples of six.
The following activity created by “The Hong Kong Academy for Gifted
Students,” can be used to teach the conceptions of prime numbers along with the conjectures of
Present the statement to pupils in either of the following ways:
“All prime numbers greater than three can be expressed in the form 6n + 1, 6n – 1,
where n is a positive whole number.”
“Every prime number greater than 3 is either one more than or one more than a multiple
of 6.”
Give the students some time to investigate the above statement (preferably using Excel)
and encourage them to come up with a proof of this proposition.
The results show that:
(i) All prime numbers so far tested (apart from a and 3) lie either in the 6n + 1 column or
the 6n – 1 column. There are no primes in the other columns.
(ii) There are, however, nonprimes in the 6n + 1 column and the 6n – 1 column.
So far all we can see is that some examples fit the conjecture. We have not yet proved
that all primes fit the conjecture. It would take forever!
Suggested proof:
Star by noticing that every whole number can be expressed in the form 6n – 2, 6n 1, 6n
+1, 6n +2, and 6n + 3. Then, notice the following facts:
(i) 6n is always divisible by 6, for all values of n (so none of the numbers in this column
can be prime).
(ii) 6n – 2 is always divisible by 2, for all values of n (so none of the numbers in this
column can be prime either, except two itself).
(iii) The same is true of 6n + 2.
(iv) 6n + 3 is always divisible by 3, for all values of n (so none of the numbers in this
column can be prime, except three itself). So, all the primes greater than three must lie in
the 6n – 1 and 6n + 1 columns. (The Hong Kong Academy for Gifted Students, 2003).
University of TennesseeMartin Professor Chris Caldwell wrote that he (along with a friend
using MATLAB (computer programming language)) found every prime number over three lies
next to a number divisible by six. After testing 1,000,000 prime numbers, take n >3 and divide it
by 6n = 6q + r where q is a nonnegative integer and the remainder r is one of 0, 1, 2, 3, 4, or 5.
•
•
If the remainder is 0, 2 or 4, then the number n is divisible by 2, and cannot be prime.
If the remainder is 3, then the number n is divisible by 3, and cannot be prime.
So if n is prime, then the remainder r is either
•
•
1 (and n = 6q + 1 is one more than a multiple of six), or
5 (and n = 6q + 5 = 6(q+1)  1 is one less than a multiple of six).
Remember that being one more or less than a multiple of six does not make a number prime. We
have only shown that all primes other than 2 and 3 (which divide 6) have this form.
Classroom Uses for Number Patterns
The hundreds chart can be used in a variety of ways in a math class. As a result,
we have noticed that the other number charts are not better for teaching students of all age simple
but complex math pattern. Furthermore, a hundreds chart helps students see patterns with
numbers. The hundreds chart can also be used to help students with many number sense related
activities, as compared to each of the four mathematical operations of addition, subtraction,
multiplication, and division. In the 10x10 hundred chart, the numbers 1100 are arranged in ten
columns and ten rows. Within the columns and rows of the 10 x 10 hundred chart, there are
several patterns in the chart, which can be identified. In looking at a 10 x 10 hundred chart, many
number patterns that are evident. Not only are there many patterns that are evident in the 10 x 10
hundred chart, but there is also a basic pattern (a formula) for the basic 10 x 10 hundred charts
(square). The number patterns that are present in a 10 x 10 hundred chart lend themselves to a
learning tool for students of all ages.
Magic squares are proving to be an ideal tool for the effective illustration of many
mathematical concepts. In fact, simple Arithmetic, which would stem from summing the
numbers in the rows, columns, and diagonals, to Algebra and Geometry with the application of
the formula mentioned in the first section of this paper (Anderson) here are many ways to
incorporate magic squares to help teach students math. Consider a few important things when
adapting magic squares for the classroom.
It is imperative to make sure the students understand a method of determining the
placement of numerals in any size magic cell – this can be done by providing students with an
example first – perhaps a 3 x 3 or 4 x 4 magic square. When providing the example, point out
how the numbers in the squares add to the same number in every direction. To make this more
concrete to the student, the instructor could provide a second example that uses the same square,
but with one or more cells missing. Students could then either find the missing consecutive
number (arithmetic) or create a formula to find the missing number that would make the “magic”
happen. For an illustration of this, see the example below.
8
3
4
16
6
1
5
9
2
10
Note: Introduce the lesson by displaying a full 3 x 3 magic
square – point out the sum of the rows, columns, and
diagonals is 15. Also introduce the formula M = n(n2 + 1)
/2 to show that the magic number is 15.
Note: Use progression by starting to block out the cells.
You can use the same number, or double or triple the
number. In this example, the numbers are doubled. Ask
students to fill in the missing numbers. They should be
able to point out that M is 30, and proceed to fill in the
missing numbers from there. Circulate and provide
guidance if necessary.
After completing this exercise with students, lead students to develop an understanding of a
method of constructing a magic square by attempting to create one of their own. Some examples
of exercises for the classroom are listed:
Exercise 1: Draw a 3 by 3 grid, and without any clues, see if the students can fill in the
numbers 1 to 9 so that the result is a magic square. If you want to give a hint; put the number
5 in the middle.
Exercise 2: Look at your final result from the last magic square; now have the students
square every number in the square. Is the square still magic? Yes! Be sure to ask students
why to ensure they understand this. Use a simple formula, like x + 3 = 8 – students will say
that x = 5. Now double the numbers – 2x + 6 = 16 – students will still say that x = 5.. Finally,
show them the formula M = n(n2 + 1)/2 and use any number for n, then ask them to square n
2
and complete the formula again. The result should be M !
Exercise 3: To begin, refer back to the first example, the 3 x 3 magic square. In this square,
the middle number is 5 and is in the center of the square. Ask students to try putting another
number in the center. After some time, it will be discovered that no other number will work
in the center position. Therefore, it can be concluded that because 5 is the median number, it
must be placed in the center position with a greater and lesser number on either side.
For this exercise, provide the students with nine prime numbers: 1, 7, 13, 31, 37, 43, 61, 67,
and 73. Can these numbers be arranged into a magic square? Be sure to remind students the
key to arranging the numbers correctly in any magic square is to realize that the middle
number (in this case, 37) must always go in the center.
67 1 43
13 37 61
31 73 7
Exercise 4: Another activity could be a 3 x 3 Magic Square: write all the number 19 on
small squares of paper and cut them out; move the numbers to the spaces so the sum of each
row, column, and main diagonal equals 15 and have the students record their work. You can
challenge them by asking if there is more than one way to place the numbers to that the sums
of each row, column, and main diagonal equals 15—have them compare with other students!
Exercise 5: A 5 x 5 Magic Square: write all the number 125 on small squares of paper and
cut them out; move the numbers to the spaces so the sum of each row, column, and main
diagonal equals 65 and have the students record their work.
The key to the mastery of this concept is building a solid foundation for students to work.
By demonstrating the mathematical concept you are trying to get students to comprehend in an
artless manner first, you are creating a bridge into the understanding of more complicated
mathematical concepts, like the magic square. Starting small and taking the time to ask why,
explain concepts, and demonstrate why the square is magic will allow you and your students to
grasp fully the magical properties of this phenomenal concept (Anderson).
Closing Remarks
Numerical patterns are just the beginning of the acknowledgement of the importance of
mathematics in one’s everyday life. Through careful observation and conjecture we have found
that numerous patterns in both number charts and magic squares. These observations can be
passed along to students beginning at an early age – both allowing them to deepen their
knowledge of the number system, and develop an awareness for patterns and puzzles in the study
of mathematics. It is our hope that the teaching of numerical patterns to elementary age children
will also develop a love of the beauty and presence of mathematics in our everyday lives.
from https://primes.utm.edu/notes/conjectures/
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