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 _{4}C_{2} = 6 ways to choose two factors of 3's, and _{3}C_{1} = 3 ways to choose one factor of 2. What is the smallest odd number that has a multiplicative partition?
2. For a given nonnegative integer n, what is the shortest unique equation for n, an equation in one variable that has unique integer solution n? We allow wellknown symbols of any kind (+, –, +, /, ^, !, √, concatenation, etc.), but we do not allow constants. For example, x + x = x has unique solution x = 0, and x^{x} = 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, 10_{89} = 108_{9}, 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 0n 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 6^{7} 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 4^{10} 22222222 1  (JD) 
80  40 10 88 44 2 1111  
84  2121 444444 2^{9}  (JD) 
88  22 4^{10} 2^{13}  (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 4^{17} 3 2^{14}  (BZ) 
110  111111 101010 5555 22 1  (BZ) 
112  28 4^{18} 22222 11  (JD) 
114  19 6 3^{28} 22 1  (BZ) 
120  30 4^{15} 2^{15}  (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.


3.
Here are the known base printer errors with 8 or fewer digits:
4Digit Printer Errors  

10_{89} = 108_{9} = 89  11_{78} = 117_{8} = 79  12_{24} = 122_{4} = 26 
12_{67} = 126_{7} = 69  13_{56} = 135_{6} = 59  14_{45} = 144_{5} = 49 
21_{89} = 218_{9} = 179  23_{78} = 237_{8} = 159  25_{67} = 256_{7} = 139 
32_{89} = 328_{9} = 269  35_{78} = 357_{8} = 239  43_{89} = 438_{9} = 359 
47_{78} = 477_{8} = 319  52_{68} = 526_{8} = 342  54_{89} = 548_{9} = 449 
65_{89} = 658_{9} = 539  76_{89} = 768_{9} = 629  87_{89} = 878_{9} = 719 
5Digit Printer Errors  

11_{110} = 111_{10} = 111  11_{889} = 1188_{9} = 890  13_{114} = 1311_{4} = 117 
13_{419} = 134_{19} = 422  14_{623} = 146_{23} = 627  15_{416} = 1541_{6} = 421 
15_{627} = 1562_{7} = 632  15_{928} = 159_{28} = 933  16_{212} = 162_{12} = 218 
18_{825} = 188_{25} = 833  23_{889} = 2388_{9} = 1781  31_{779} = 3177_{9} = 2338 
34_{214} = 342_{14} = 646  35_{889} = 3588_{9} = 2672  36_{624} = 366_{24} = 1878 
44_{420} = 444_{20} = 1684  47_{522} = 375_{22} = 2095  47_{889} = 4788_{9} = 3563 
99_{930} = 999_{30} = 8379 
6Digit Printer Errors  

10_{1010} = 1010_{10} = 1010  10_{2537} = 10253_{7} = 2537  11_{1110} = 1111_{10} = 1111 
12_{1210} = 1212_{10} = 1212  12_{7719} = 1277_{19} = 7721  13_{1310} = 1313_{10} = 1313 
13_{3617} = 13361_{7} = 3620  13_{6918} = 1369_{18} = 6921  14_{1410} = 1414_{10} = 1414 
15_{1510} = 1515_{10} = 1515  15_{8819} = 1588_{19} = 8824  16_{1610} = 1616_{10} = 1616 
17_{1710} = 1717_{10} = 1717  18_{1810} = 1818_{10} = 1818  18_{9919} = 1899_{19} = 9927 
19_{1910} = 1919_{10} = 1919  20_{2467} = 20246_{7} = 4934  22_{1546} = 22154_{6} = 3094 
29_{4415} = 2944_{15} = 8839  42_{2607} = 42260_{7} = 10430  45_{1912} = 4519_{12} = 7653 
80_{8020} = 8080_{20} = 64160 
7Digit Printer Errors  

10_{34608} = 103460_{8} = 34608  11_{11110} = 11111_{10} = 11111  24_{42048} = 244204_{8} = 84100 
8Digit Printer Errors  

10_{101010} = 101010_{10} = 101010  11_{111110} = 111111_{10} = 111111  12_{121210} = 121212_{10} = 121212 
13_{131310} = 131313_{10} = 131313  13_{670914} = 136709_{14} = 670917  14_{141410} = 141414_{10} = 141414 
15_{151510} = 151515_{10} = 151515  15_{358512} = 153585_{12} = 358517  16_{161610} = 161616_{10} = 161616 
17_{171710} = 171717_{10} = 171717  18_{181810} = 181818_{10} = 181818  19_{191910} = 191919_{10} = 191919 
24_{334748} = 2433474_{8} = 669500  38_{704329} = 3870432_{9} = 2112995 
Joe DeVincentis found these 9 digit printer errors:
9Digit Printer Errors  

11_{1111110} = 1111111_{10} = 1111111  12_{6214349} = 12621434_{9} = 6214351  14_{3255108} = 14325510_{8} = 3255112 
15_{4318512} = 1543185_{12} = 4318517  34_{5603179} = 34560317_{9} = 16809541  62_{4973113} = 6249731_{13} = 29838680 
Joe DeVincentis noticed that all but one of the 4digit solutions are of the form ABCD with C=D1 and B=(10D)A1.
Joe DeVincentis found two infinite families of solutions: 1N_{1N1N...1N10} = 1N1N...1N_{10} and 11_{11...10} = 11...1_{10}.
4.
Here are the largest known nonlicense plate numbers using the digits 0n:
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 = 5^{2}  
2  279,936 = 6^{7} = (9–3)^{9–2}  
3  31,381,059,609 = 9^{11} = 9^{5+6} = 3^{30–8+0}  (JD) 
4  1,125,899,906,842,624 = 2^{50} = (4×8)^{9+1} = (4×8)^{9+1} = 2^{9×6–6+2}  (JD) 
5  1,152,921,504,606,846,976 = 2^{60} = 2^{60} = 4^{5×6} = 4^{5×6} = 8^{17+1+1+9/9}  (JD) 
6  4,722,366,482,869,645,213,696 = 2^{72} = 4^{36} = 4^{36} = 8^{24} = 8^{5×6–(6+6)/2} = 16^{9+9}  (JD) 
7  42,391,158,275,216,203,514,294,433,201 = 3^{60} = 3^{3×4×5} = (2+1)^{59+1} = 9^{30} = 27^{4×5} = (24+4–1)^{22–2} = 81^{15}  (JD) 
8  324,518,553,658,426,726,783,156,020,576,256 = 4^{54} = 8^{36} = 8^{36} = 8^{36} = (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 = 3^{72} = 3^{72} = (41)^{72} = (41)^{76–4} = 9^{9×4} = 9^{40–4} = 9^{8×4+4} = 27^{5×5–1} = 81^{18}  (JD) 
10  147,808,829,414,345,923,316,083,210,206,383,297,601 = 3^{80} = 3^{80} = 3^{80} = 3^{80} = 3^{72+8} = 3^{74+6} = 9^{40} = 9^{46–6} = 9^{4×5×2×1} = (11–2)^{(21–1)×2}  (JD) 
11  1,427,247,692,705,959,881,058,285,969,449,495,136,382,746,624
= 4^{75} = 4^{75} = 4^{75} = 8^{50} = 8^{50} = 4^{73+2} = 4^{81–6} = 4^{81–6} = 4^{69+6} = 2^{122+6} = 32^{9+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
= 4^{78} = 4^{78} = 4^{78} = 4^{78} = 4^{78} = 8^{52} = 8^{52} = 16^{39} = 16^{39} = 16^{39} = 2^{300/2+3+3} = (73)^{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 = 5^{6} = 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
= (20–4^{0})! = (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)!–8^{0})! = (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+600000^{0})!  (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+60^{0})! = (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 email me. Click here to go back to Math Magic. Last updated 6/26/10.