1. Which positive integers n have a multiplicative partition, a partition with the property that there are exactly n subsets of the partition with product n, including one that contains any given summand? For example, 18 = 3 + 3 + 3 + 3 + 2 + 2 + 2 has this property, since there are 4C2 = 6 ways to choose two factors of 3's, and 3C1 = 3 ways to choose one factor of 2. What is the smallest odd number that has a multiplicative partition?
2. For a given non-negative integer n, what is the shortest unique equation for n, an equation in one variable that has unique integer solution n? We allow well-known symbols of any kind (+, –, ×, /, ^, !, √, concatenation, etc.), but we do not allow constants. For example, x + x = x has unique solution x = 0, and xx = x has unique solution x = 1. What are the equations using the fewest symbols that uniquely identify the positive integers?
3. In June 2001, we investigated printer's errors involving powers. But if a subscript is used to denote a base, there are also strings of digits that evaluate to the same number for different sets of subscripted digits. We call these base printer errors. For example, 1089 = 1089, as both evaluate to 89. Can you find some other base printer errors? What infinite families are there? Are there strings that can be interpreted 3 different ways to give the same result?
4. On long car trips, we play a game with numerical license plates. We try to insert symbols between the digits to make a true equation. For example, 4935 has solutions √4 + 9/3 = 5 and √4 + 9 = 3! + 5. Thus we say the number 4935 is a license plate number. What is the largest number that is not a license plate number? This may be too hard, so what is the largest number only using the digits 0-n that is not a license plate number?
5. In August 2000, we investigated Friedman numbers. But some numbers have the ability to make themselves multiple times using their own digits. For example, using each of the digits of 279936 exactly once, we can form 67 and (9–3)9–2, both of which evaluate to 279936. Is this the smallest multiple Friedman number? What is the smallest Friedman number of order 3? of order n?
1.
Bryce Herdt showed that prime numbers, and products of two prime numbers, do not have multiplicative partitions.
Joe DeVincentis proved that 105 was the smallest odd number with a multiplicative partition. He also showed that 32 is the smallest power of 2 that has a multiplicative partition.
Joe DeVincentis showed that 4p, where p is an odd prime, has a multiplicative partition when p=5 or (5p+3)/2 is a square. Berend van der Zwaag showed that when p>16 and (7p+1)/2 is square also works.
Here are the known numbers with multiplicative partitions:
| n | Partition | Author |
|---|---|---|
| 1 | 1 | |
| 12 | 3 2222 1 | |
| 18 | 3333 222 | |
| 20 | 55 22222 | |
| 24 | 66 4 3 2 111 | |
| 32 | 16 44 222 11 | (BH) |
| 36 | 6666 333 2 1 | (BH) |
| 40 | 10 5555 4 22 11 | |
| 42 | 77 6 3333 22222 | (BH) |
| 48 | 16 8 6 33333 111 | |
| 54 | 66666 3333 222222 | (BZ) |
| 56 | 1414 8 7 44 2 111 | |
| 60 | 1515 5 44444 3 11 | |
| 64 | 32 16 22222222 | (JD) |
| 66 | 111111 6 3333333 222 | (BH) |
| 72 | 36 1212 3 222 111 | |
| 76 | 19 410 22222222 1 | (JD) |
| 80 | 40 10 88 44 2 1111 | |
| 84 | 2121 444444 29 | (JD) |
| 88 | 22 410 213 | (JD) |
| 90 | 1010 999 5555555 2222 | (BH) |
| 92 | 46 23 4 222222222 1 | (BZ) |
| 96 | 48 16 6666 22 1111 | (BH) |
| 100 | 2525 555 4444444 222 1 | (BH) |
| 102 | 171717 6666 33333 222222 | (BZ) |
| 104 | 26 131313 88 4444 222 1 | (BZ) |
| 105 | 15 7777777 5555555 33 | (JD) |
| 108 | 9 417 3 214 | (BZ) |
| 110 | 111111 101010 5555 22 1 | (BZ) |
| 112 | 28 418 22222 11 | (JD) |
| 114 | 19 6 328 22 1 | (BZ) |
| 120 | 30 415 215 | (JD) |
2.
Below are the shortest known equations. We count square brackets [ ] but not round parentheses ( ), which just extend the square root, due to my poor typesetting skills. Bryce Herdt convinced me that since concatenation was only allowed between x's, that in many cases a space could mean multiplication. Gavin Theobald convinced me that subscripts could be used for bases.
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3.
Here are the known base printer errors with 8 or fewer digits:
| 4-Digit Printer Errors | |||||
|---|---|---|---|---|---|
| 1089 = 1089 = 89 | 1178 = 1178 = 79 | 1224 = 1224 = 26 | 1267 = 1267 = 69 | 1356 = 1356 = 59 | 1445 = 1445 = 49 |
| 2189 = 2189 = 179 | 2378 = 2378 = 159 | 2567 = 2567 = 139 | 3289 = 3289 = 269 | 3578 = 3578 = 239 | 4389 = 4389 = 359 |
| 4778 = 4778 = 319 | 5268 = 5268 = 342 | 5489 = 5489 = 449 | 6589 = 6589 = 539 | 7689 = 7689 = 629 | 8789 = 8789 = 719 |
| 5-Digit Printer Errors | ||||
|---|---|---|---|---|
| 11110 = 11110 = 111 | 11889 = 11889 = 890 | 13114 = 13114 = 117 | 13419 = 13419 = 422 | 14623 = 14623 = 627 |
| 15416 = 15416 = 421 | 15627 = 15627 = 632 | 15928 = 15928 = 933 | 16212 = 16212 = 218 | 18825 = 18825 = 833 |
| 23889 = 23889 = 1781 | 31779 = 31779 = 2338 | 34214 = 34214 = 646 | 35889 = 35889 = 2672 | 36624 = 36624 = 1878 |
| 44420 = 44420 = 1684 | 47522 = 37522 = 2095 | 47889 = 47889 = 3563 | 99930 = 99930 = 8379 |
| 6-Digit Printer Errors | |||
|---|---|---|---|
| 101010 = 101010 = 1010 | 102537 = 102537 = 2537 | 111110 = 111110 = 1111 | 121210 = 121210 = 1212 |
| 127719 = 127719 = 7721 | 131310 = 131310 = 1313 | 133617 = 133617 = 3620 | 136918 = 136918 = 6921 |
| 141410 = 141410 = 1414 | 151510 = 151510 = 1515 | 158819 = 158819 = 8824 | 161610 = 161610 = 1616 |
| 171710 = 171710 = 1717 | 181810 = 181810 = 1818 | 189919 = 189919 = 9927 | 191910 = 191910 = 1919 |
| 202467 = 202467 = 4934 | 221546 = 221546 = 3094 | 294415 = 294415 = 8839 | 422607 = 422607 = 10430 |
| 451912 = 451912 = 7653 | 808020 = 808020 = 64160 |
| 7-Digit Printer Errors | ||
|---|---|---|
| 1034608 = 1034608 = 34608 | 1111110 = 1111110 = 11111 | 2442048 = 2442048 = 84100 |
| 8-Digit Printer Errors | ||
|---|---|---|
| 10101010 = 10101010 = 101010 | 11111110 = 11111110 = 111111 | 12121210 = 12121210 = 121212 |
| 13131310 = 13131310 = 131313 | 13670914 = 13670914 = 670917 | 14141410 = 14141410 = 141414 |
| 15151510 = 15151510 = 151515 | 15358512 = 15358512 = 358517 | 16161610 = 16161610 = 161616 |
| 17171710 = 17171710 = 171717 | 18181810 = 18181810 = 181818 | 19191910 = 19191910 = 191919 |
| 24334748 = 24334748 = 669500 | 38704329 = 38704329 = 2112995 |
Joe DeVincentis found these 9-digit printer errors:
| 9-Digit Printer Errors | ||
|---|---|---|
| 111111110 = 111111110 = 1111111 | 126214349 = 126214349 = 6214351 | 143255108 = 143255108 = 3255112 |
| 154318512 = 154318512 = 4318517 | 345603179 = 345603179 = 16809541 | 624973113 = 624973113 = 29838680 |
Michael Yee found these 10-digit printer errors:
| 10-Digit Printer Errors | ||
|---|---|---|
| 1360625549 = 1360625549 = 60625552 | 2017023608 = 2017023608 = 34047216 | 2146701249 = 2146701249 = 93402499 |
| 5447386119 = 5447386119 = 236930599 | 6648265709 = 6648265709 = 289594260 | 1010101010 = 1010101010 = 10101010 |
| 1111111110 = 1111111110 = 11111111 | 1212121210 = 1212121210 = 12121212 | 1313131310 = 1313131310 = 13131313 |
| 1414141410 = 1414141410 = 14141414 | 1515151510 = 1515151510 = 15151515 | 1616161610 = 1616161610 = 16161616 |
| 1717171710 = 1717171710 = 17171717 | 1818181810 = 1818181810 = 18181818 | 1919191910 = 1919191910 = 19191919 |
Joe DeVincentis noticed that all but one of the 4-digit solutions are of the form ABCD with C=D-1 and B=(10–D)A–1.
Joe DeVincentis found two infinite families of solutions: 1N1N1N...1N10 = 1N1N...1N10 and 1111...10 = 11...110.
4.
Here are the largest known non-license plate numbers using the digits 0-n:
| n | Largest | Author |
|---|---|---|
| 0 | 0 | |
| 1 | 1 | (BH) |
| 2 | 21 | |
| 3 | 32 | (BZ) |
| 4 | 43 | (BZ) |
| 5 | 553 | (BZ) |
| 6 | 655 | (BZ) |
| 7 | 7662 | (BZ) |
| 8 | 8775 | (BZ) |
| 9 | 8775 | (BZ) |
5.
Here are the smallest known Friedman numbers of order n:
| n | Smallest Known Friedman Number of Order n | Author |
|---|---|---|
| 1 | 25 = 52 | |
| 2 | 279,936 = 67 = (9–3)9–2 | |
| 3 | 31,381,059,609 = 911 = 95+6 = 330–8+0 | (JD) |
| 4 | 1,125,899,906,842,624 = 250 = (4×8)9+1 = (4×8)9+1 = 29×6–6+2 | (JD) |
| 5 | 1,152,921,504,606,846,976 = 260 = 260 = 45×6 = 45×6 = 817+1+1+9/9 | (JD) |
| 6 | 4,722,366,482,869,645,213,696 = 272 = 436 = 436 = 824 = 85×6–(6+6)/2 = 169+9 | (JD) |
| 7 | 42,391,158,275,216,203,514,294,433,201 = 360 = 33×4×5 = (2+1)59+1 = 930 = 274×5 = (24+4–1)22–2 = 8115 | (JD) |
| 8 | 324,518,553,658,426,726,783,156,020,576,256 = 454 = 836 = 836 = 836 = (7+1)62 = (2+2)55–1 = (5+5+6)27 = (5+5+6)27+0+0 | (JD) |
| 9 | 22,528,399,544,939,174,411,840,147,874,772,641 = 372 = 372 = (4-1)72 = (4-1)76–4 = 99×4 = 940–4 = 98×4+4 = 275×5–1 = 8118 | (JD) |
| 10 | 147,808,829,414,345,923,316,083,210,206,383,297,601 = 380 = 380 = 380 = 380 = 372+8 = 374+6 = 940 = 946–6 = 94×5×2×1 = (11–2)2(21–1) | (JD) |
| 11 | 1,427,247,692,705,959,881,058,285,969,449,495,136,382,746,624
= 475 = 475 = 475 = 850 = 850 = 473+2 = 481–6 = 481–6 = 469+6 = 2122+6 = 329+9+9+9+9+2–9–8 | (JD) |
| 12 | 91,343,852,333,181,432,387,730,302,044,767,688,728,495,783,936
= 478 = 478 = 478 = 478 = 478 = 852 = 852 = 1639 = 1639 = 1639 = 2300/2+3+3 = (7–3)3×(30–3)–3 | (JD) |
Joe DeVincentis suggested that factorials should be allowed. Here are the best known solutions in this case:
| n | Smallest Known Factorial Friedman Number of Order n | Author |
|---|---|---|
| 1 | 1! = 1 | (BZ) |
| 2 | 15,625 = 56 = 5(1+2)! | (BH) |
| 3 | 479,001,600 = (19–7)! = (4!/(0!+0!))! = (6×(0!+0!))! | (BZ) |
| 4 | 1,307,674,368,000 = (14+0!)! = (7+8)! = (3×(6–0!))! = (7+6+3–0!+0)! | (JD) |
| 5 | 20,922,789,888,000 = (8×2)! = (8×2)! = (8×2)! = (9+7)! = (8×(0!+0!)+0×90)! | (JD) |
| 6 | 121,645,100,408,832,000
= (20–40)! = (18+0!)! = (18+0!)! = (4!–5)! = (3×6+0!)! = (20–1)! | (BZ) |
| 7 | 2,432,902,008,176,640,000 = 20! = 20! = 20! = (19+0!)! = (3×7–0!)! = (8+6+6+0+0)! = (4!–4)! | |
| 8 | 25,852,016,738,884,976,640,000
= 23! = (4!–0!)! = (4!–0!)! = ((8/2)!–80)! = (5×6–7)! = (5×6–7)! = (9+8+6)! = ((8/(0!+0!))!–1)! | (BZ) |
| 9 | 620,448,401,733,239,439,360,000 = 4!! = 4!! = 4!! = 4!! = (3×8)! = ((1+7)×3)! = (9×3–3)! = (2×9+6)! = (23+6000000)! | (JD) |
| 10 | 620,448,401,733,239,439,360,000 = 4!! = 4!! = 4!! = 4!! = (3×8)! = ((1+7)×3)! = (9×3–3)! = (2×9+6)! = (23+600)! = (0!+0!+0!+0!)!! | |
| 11 | 620,448,401,733,239,439,360,000 = 4!! = 4!! = 4!! = 4!! = (3×8)!= ((1+7)×3)! = (9×3–3)! = (2×9+6)! = (6–2+0)!! = (3+0!)!! = (0!+0!+0!+0!)!! | (BH) |
| 12 | 620,448,401,733,239,439,360,000 = 4!! = 4!! = 4!! = 4!! = (3×8)! = (1+3)!! = (7–3)!! = (6–2)!! = (6–2)!! = (3+0!)!! = (3+0!)!! = (0!+0!+0!+0!+9–9)!! | (JD) |
If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 6/26/10.