Problem of the Month (May 2010)

This month we consider several problems concerning numbers:

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?

This month's contributors were Jaroslaw Wroblewski, Bryce Herdt, Joe DeVincentis, Berend van der Zwaag, and Gavin Theobald.

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:

Known Numbers with
Multiplicative Partitions
nPartitionAuthor
11
123   2222   1
183333   222
2055   22222
2466   4   3   2   111
3216   44   222   11(BH)
366666   333   2   1(BH)
4010   5555   4   22  11
4277   6   3333   22222(BH)
4816   8   6   33333   111
5466666   3333   222222(BZ)
561414   8   7   44   2   111
601515   5   44444   3   11
6432   16   22222222(JD)
66111111   6   3333333   222(BH)
7236   1212   3   222   111
7619   410   22222222   1(JD)
8040   10   88   44   2   1111
842121   444444   29(JD)
8822   410   213(JD)
901010   999   5555555   2222(BH)
9246   23   4   222222222   1(BZ)
9648   16   6666   22   1111(BH)
1002525   555   4444444   222   1(BH)
102171717   6666   33333   222222(BZ)
10426   131313   88   4444   222   1(BZ)
10515   7777777   5555555   33(JD)
1089   417   3   214(BZ)
110111111   101010   5555   22   1(BZ)
11228   418   22222   11(JD)
11419   6   328   22   1(BZ)
12030   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.

Shortest Known Equations with Unique Solution n
nEquationSymbolsAuthor
0x + x = x5
1xx = x4
2xx = x + x6
3x! = x + x6
4√(x! – x – x) = x 9
5x! = x · x · x – x10
6x! = xxx + xx – x – x13(BH)
7√(x √(xxx + x)) = x + x12(GT)
8x · x / √(x+x) = x + x12(GT)
9√x + √x = √(x)!9(GT)
10xx = x4(GT)
11xx = x + x/x8(GT)
12x[xx – x] = x + x11(GT)
13x √√xx = x + x9(GT)
14x √(xx – x) = x + x10(GT)
15x √√(x + x/x) = x + x12(BZ)
16x √(x/√x) = x + x10(GT)
17xx = √(x – x/x)!10(JD)
18√√(x+x) = x / √(x + x + x)14(JD)
19x √(xx – x) = x + x + x12(GT)
20xx = x + x6(GT)
21xx = x + x + x/x10(GT)
22x[xx – x – x] = x + x13(GT)
23x[xx – x – x] = x + x + x15(GT)
24x √(xx – x – x) = x + x12(GT)
25xx = x + x + √x9(GT)
26xx = x + x + √(x – x/x) + x/x17(GT)
27x · x / √(x + x + x) = x + x + x16(BZ)
28xx = x + x + √(xx)10(GT)
29x √(xx – x – x) = x + x + x14(GT)
30xx = x + x + x8(GT)
31xx = x + x + x + x/x12(GT)
32x √(x / √(x + x)) = x + x12(GT)
33xx · x – xx = x9(GT)
34x √(xx – x – x – x) = x + x14(GT)
35x √(xx – xx/x) = x + x + x15(GT)
36√(x)! = xx √x + x10(GT)
nEquationSymbolsAuthor
37x√√(xx – x) = x + x + x13(GT)
38x √(xx – x – x – x + x/x) = x + x + x20(GT)
39x √(xx – x – x – x) = x + x + x16(GT)
40√(x · xx) = x + x9(GT)
41√x √(xx – x/x) = x + x13(GT)
42√(xx · x – x– x) = x + x13(GT)
43√(xx · x – x– x – x) = x + x15(GT)
44x √(xx) / √(x + x/x) = x + x15(GT)
45√(xx · x – x– x – x – x – x) = x + x19(GT)
46xx + x[x + x – xx] = x + x + x18(GT)
47xx/x + x + x = xx11(GT)
48x · xx = xx + xx – x – x14(JD)
49xx + x + x = √(x)! + √x13(JW)
50√x √(xx – x) = x + x11(GT)
51x √√√(xx) = x + x10(GT)
52√(x[xx – x] – x – x) = x + x16(GT)
53√(x[xx – x] – x – x – x) = x + x17(GT)
54x √(xx – x) / √(x + x/x) = x + x17(GT)
55x √(xx / [x + x/x] – x/x) = x + x20(GT)
59x[xx · x – x]/xx = x + x + x18(GT)
60[xx / x]! = xx + xx13(GT)
64√√(√x + √x) = [x + x]/x15(JD)
66x √(xx · x + x + x)/√xx = x + x18(GT)
70√x √(xx + x + x) = x + x + x15(GT)
71√x √(xx + x + x – x/x) = x + x + x19(GT)
72x √(x √(x+x)) / √(x + x + x) = x + x18(GT)
80x √xx / √(x + x) = x + x13(GT)
81x √(x/√x) = x + x + x12(GT)
82x √√(x – x/x) = x + x + x14(GT)
83x √√(x – [x + x]/x) = x + x + x18(GT)
84x √√(x – [x + x + x]/x) = x + x + x20(GT)
89xx – x/x = [[x + x + x]/x]!!20(GT)
90√(x · xx) = x + x + x11(GT)
91√(xx · x – x) = x + x + x13(GT)
92√(xx · x – x – x) = x + x + x15(GT)
93√(xx · x – x – x – x) = x + x + x17(GT)
94√(xx · x – x – x – x – x) = x + x + x19(GT)
99xxx / xx = x + x/x12(GT)
100√(xx) = x5(JD)

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

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:

Largest Known
Plate Numbers
nLargestAuthor
00
11(BH)
221
332(BZ)
443(BZ)
5553(BZ)
6655(BZ)
77662(BZ)
88775(BZ)
98775(BZ)

5.

Here are the smallest known Friedman numbers of order n:

Smallest Known Multiple Friedman Numbers
nSmallest Known Friedman Number of Order nAuthor
125 = 52
2279,936 = 67 = (9–3)9–2
331,381,059,609 = 911 = 95+6 = 330–8+0(JD)
41,125,899,906,842,624
= 250 = (4×8)9+1 = (4×8)9+1 = 29×6–6+2
(JD)
51,152,921,504,606,846,976
= 260 = 260 = 45×6 = 45×6 = 817+1+1+9/9
(JD)
64,722,366,482,869,645,213,696
= 272 = 436 = 436 = 824 = 85×6–(6+6)/2 = 169+9
(JD)
742,391,158,275,216,203,514,294,433,201
= 360 = 33×4×5 = (2+1)59+1 = 930
= 274×5 = (24+4–1)22–2 = 8115
(JD)
8324,518,553,658,426,726,783,156,020,576,256
= 454 = 836 = 836 = 836 = (7+1)62
= (2+2)55–1 = (5+5+6)27 = (5+5+6)27+0+0
(JD)
922,528,399,544,939,174,411,840,147,874,772,641
= 372 = 372 = (4-1)72 = (4-1)76–4 = 99×4
= 940–4 = 98×4+4 = 275×5–1 = 8118
(JD)
10147,808,829,414,345,923,316,083,210,206,383,297,601
= 380 = 380 = 380 = 380 = 372+8 = 374+6
= 940 = 946–6 = 94×5×2×1 = (11–2)(21–1)×2
(JD)
111,427,247,692,705,959,881,058,285,969,449,495,136,382,746,624
= 475 = 475 = 475 = 850 = 850 = 473+2 = 481–6
= 481–6 = 469+6 = 2122+6 = 329+9+9+9+9+2–9–8
(JD)
1291,343,852,333,181,432,387,730,302,044,767,688,728,495,783,936
= 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:

Smallest Known Multiple Factorial Friedman Numbers
nSmallest Known Factorial Friedman Number of Order nAuthor
11! = 1(BZ)
215,625 = 56 = 5(1+2)!(BH)
3479,001,600 = (19–7)! = (4!/(0!+0!))! = (6×(0!+0!))!(BZ)
41,307,674,368,000
= (14+0!)! = (7+8)! = (3×(6–0!))! = (7+6+3–0!+0)!
(JD)
520,922,789,888,000
= (8×2)! = (8×2)! = (8×2)! = (9+7)! = (8×(0!+0!)+0×90)!
(JD)
6121,645,100,408,832,000
= (20–40)! = (18+0!)! = (18+0!)!
= (4!–5)! = (3×6+0!)! = (20–1)!
(BZ)
72,432,902,008,176,640,000
= 20! = 20! = 20! = (19+0!)!
= (3×7–0!)! = (8+6+6+0+0)! = (4!–4)!

825,852,016,738,884,976,640,000
= 23! = (4!–0!)! = (4!–0!)! = ((8/2)!–80)! = (5×6–7)!
= (5×6–7)! = (9+8+6)! = ((8/(0!+0!))!–1)!
(BZ)
9620,448,401,733,239,439,360,000
= 4!! = 4!! = 4!! = 4!! = (3×8)! = ((1+7)×3)!
= (9×3–3)! = (2×9+6)! = (23+6000000)!
(JD)
10620,448,401,733,239,439,360,000
= 4!! = 4!! = 4!! = 4!! = (3×8)! = ((1+7)×3)!
= (9×3–3)! = (2×9+6)! = (23+600)! = (0!+0!+0!+0!)!!
11620,448,401,733,239,439,360,000
= 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)
12620,448,401,733,239,439,360,000
= 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.