# Problem of the Month (August 2003)

The number 3252 has the property that it can be written with its digits in order using only plus signs and a squaring operator:

3252 = 3+((22)2+52+(22)2)2

Since the exponents get out of hand, we can leave them off and let parentheses mean "square it":

3252 = 3+(((2))+(5)+((2)))

Here are the numbers with 2 or 3 digits that can be written like this: 81, 89, 136, 264, 333, 337, 425, 441, 445. Can you find the appropriate squaring expressions? How many 4-digit numbers are there with this squaring property? Do all large numbers have this property?

Call numbers with the above squaring property S-numbers.

Joseph DeVincentis, Don Rogers, Trevor Green, Philippe Fondanaiche, Gyozo Nagy, and Su Jianning verified the 2 and 3 digit S-numbers.

Joseph DeVincentis, Don Rogers, and Trevor Green also found that powers of 10 could not be S-numbers. Joseph DeVincentis, Trevor Green, and Mark Michell found two other classes that cannot be S-numbers: odd numbers containing an even number of odd digits, and even numbers containing an odd number of odd digits.

Trevor Green showed that there are infinitely many S-numbers in base 8: 222k+1. He conjectures that the density of such numbers is 0 though in every base. Philippe Fondanaiche showed that there are infinitely may S-numbers in base 10: 100...0036 =(((1+0+0+...+0+0+(3)))) + (6).

Philippe Fondanaiche, Gyozo Nagy, and Su Jianning sent all the 4-digit S-numbers. Bryan Boswell sent all but one. Philippe Fondanaiche sent solutions for cubes, and for multiplication as well. Mark Michell extended the tables for multiplication and for other bases. Stewart Robert Hinsley extended the results for cubes and higher powers.

Gyozo Nagy sent 452 5-digit S-numbers, but said there are a few more he hasn't found yet. Mark Michell and Stewart Robert Hinsley found all 528 5-digit S numbers.

Here are the S-numbers with 4 or fewer digits:

 81 = (8+1) 89 = 8+(9) 136 = (1+(3))+(6) 264 = 2+6+((4)) 333 = (3)+((3)+(3)) 337= ((3))+((3)+7) 425 = (4+((2)))+(5) 441 = (4+(4)+1) 445 = 4+((4)+5) 1250 = ((1+(2)))+((5))+0 1251 = ((1+(2)))+((5))+1 1252 = ((1+(2)))+((5))+2 1253 = ((1+(2)))+((5))+3 1254 = ((1+(2)))+((5))+4 1255 = ((1+(2)))+((5))+5 1256 = ((1+(2)))+((5))+6 1257 = ((1+(2)))+((5))+7 1258 = ((1+(2)))+((5))+8 1259 = ((1+(2)))+((5))+9 1306 = 1+(3)+0+((6)) 1346 = 1+(3+4)+((6)) 1386 = 1+((3))+8+((6)) 1521 = ((1+5)+2+1) 1525 = (1+(5)+(2))+((5)) 1634 = (1+(6))+(3)+((4)) 1636 = (1+(6)+3)+(6) 1674 = (1+(6))+(7)+((4)) 1693 = (1+(6))+(9+(3)) 1745 = (1+7)+((4)+(5)) 2025 = (((2))+0+(2)+(5)) 2116 = ((2+1)+1+(6)) 2225 = ((2+(2))+(2))+((5)) 2401 = ((2+4+0+1)) 2403 = 2+((4+0+3)) 2407 = 2+4+0+((7)) 2465 = (((2)))+((4)+6+(5)) 2536 = (((2))+(5)+(3))+(6) 2576 = (((2)))+((5)+7)+((6)) 2601 = (2+(6+0+1)) 2624 = ((2)+(6))+(((2))+(4)) 2713 = (2+(7)+1)+(3) 3252 = 3+(((2))+(5)+((2))) 3258 = (3)+((2+5)+8) 3364 = ((3+3)+6+(4)) 3367 = 3+(3+6+(7)) 3706 = (3)+((7))+0+((6)) 3721 = (3+(7)+(2+1)) 3786 = ((3))+((7))+8+((6)) 3969 = ((3)+9+(6)+9) 4112 = ((4+(1+1)))+((2)) 4225 = ((4+2)+(2)+(5)) 4241 = (4)+(((2)+4)+1) 4356 = ((4)+(3)+5+(6)) 5381 = ((5)+(3))+((8)+1) 5475 = ((5))+((4)+(7))+((5)) 5785 = ((5)+7)+((8)+5) 6572 = 6+5+((7+2)) 6579 = 6+5+7+((9)) 6603 = 6+(6)+0+(((3))) 6649 = (6)+(6)+(4)+((9)) 6669 = (6)+(6)+(6)+((9)) 6730 = (6+7)+(((3)))+0 6731 =(6+7)+(((3)))+1 6732 =(6+7)+(((3)))+2 6733 =(6+7)+(((3)))+3 6734 =(6+7)+(((3)))+4 6735 =(6+7)+(((3)))+5 6736 =(6+7)+(((3)))+6 6737 =(6+7)+(((3)))+7 6738 =(6+7)+(((3)))+8 6739 =(6+7)+(((3)))+9 6779 = (6+7)+(7)+((9)) 6934 = (6)+(9)+(((3)))+((4)) 7232 = 7+(2+((3))+2) 7234 = 7+2+(((3))+4) 7274 = (7)+((2+7)+4) 7276 = (7)+2+((7)+(6)) 7328 = ((7)+3)+((2)+(8)) 7618 = (7)+(6+(1+8)) 7634 = (7)+(6+((3)))+(4) 7699 = (7)+(6+(9))+(9) 7744 = (7+(7)+(4)+(4)) 7906 = (7)+((9))+0+((6)) 7928 = 7+(9+((2))+(8)) 7969 = (7+(9))+(6+9) 8136 = ((8+1)+(3))+(6) 8208 = ((8))+((2))+0+((8)) 8228 = ((8))+(2+(2))+((8)) 8281 = (8+2+(8+1)) 8289 = 8+(2+8+(9)) 8970 = 8+((9))+((7))+0 8971 = 8+((9))+((7))+1 8972 = 8+((9))+((7))+2 8973 = 8+((9))+((7))+3 8974 = 8+((9))+((7))+4 8975 = 8+((9))+((7))+5 8976 = 8+((9))+((7))+6 8977 = 8+((9))+((7))+7 8978 = 8+((9))+((7))+8 8979 = 8+((9))+((7))+9 9125 = ((9)+1)+((2+5)) 9127 = ((9)+1)+2+((7)) 9162 = ((9))+((1+6)+2) 9216 = (9+((2+1))+6) 9346 = (9)+(((3)))+((4)+(6)) 9410 = ((9)+(4))+1+0 9411 = ((9)+(4))+1+1 9412 = ((9)+(4))+1+2 9413 = ((9)+(4))+1+3 9414 = ((9)+(4))+1+4 9415 = ((9)+(4))+1+5 9416 = ((9)+(4))+1+6 9417 = ((9)+(4))+1+7 9418 = ((9)+(4))+1+8 9419 = ((9)+(4))+1+9 9434 = 9+(4)+(((3))+(4)) 9474 = ((9)+(4))+(7)+(4) 9490 = ((9)+(4))+(9)+0 9491 = ((9)+(4))+(9)+1 9492 = ((9)+(4))+(9)+2 9493 = ((9)+(4))+(9)+3 9494 = ((9)+(4))+(9)+4 9495 = ((9)+(4))+(9)+5 9496 = ((9)+(4))+(9)+6 9497 = ((9)+(4))+(9)+7 9498 = ((9)+(4))+(9)+8 9499 = 9+((4)+(9))+(9) 9634 = (9+6)+(((3))+(4)) 9769 = ((9)+7)+((6)+9) 9925 = ((9))+(9+(2+5))

 153 = 1+(5)+(3) 224 = (2)+(2+4) 226 = 2+(2)+(6) 243 = (2+4)+(3) 370 = (3)+(7)+0 371 = (3)+(7)+1 372 = (3)+(7)+2 373 = (3)+(7)+3 374 = (3)+(7)+4 375 = (3)+(7)+5 376 = (3)+(7)+6 377 = (3)+(7)+7 378 = (3)+(7)+8 379 = (3)+(7)+9 407 = (4)+0+(7) 512 = (5+1+2) 517 = 5+(1+7) 518 = 5+1+(8) 736 = 7+(3+6) 739 = 7+3+(9) 1125 = ((1+1)+2)+(5) 1216 = (1+(2)+1)+(6) 1456 = (1+4)+(5+6) 1459 = 1+(4+5)+(9) 1547 = (1+5)+(4+7) 1674 = (1+6)+(7+4) 1729 = (1+7+2)+(9) 2205 = (2)+((2)+0+5) 2240 = ((2))+((2)+4)+0 2241 = ((2))+((2)+4)+1 2242 = ((2))+((2)+4)+2 2243 = ((2))+((2)+4)+3 2244 = ((2))+((2)+4)+4 2245 = ((2))+((2)+4)+5 2246 = ((2))+((2)+4)+6 2247 = ((2))+((2)+4)+7 2248 = ((2))+((2)+4)+8 2249 = ((2))+((2)+4)+9 2752 = (2+7+5)+(2) 2759 = (2)+7+(5+9) 2773 = 2+(7+7)+(3) 3259 = 3+((2))+(5+9) 3527 = (3)+(5)+((2)+7) 3925 = (3+9)+((2)+5) 4224 = (4)+((2)+(2))+(4) 4225 = 4+((2)+(2))+(5) 4285 = (4)+((2)+8)+(5) 4829 = 4+(8+(2))+(9) 4889 = (4)+(8+8)+(9) 4913 = (4+9+1+3) 4917 = 4+(9+1+7) 5129 = (5+1)+((2)+9) 5832 = (5+8+3+2) 5837 = 5+(8+3+7) 6865 = 6+(8+6+5) 9928 = (9+9)+((2)+8) 12167 = (1+(2)+1+6+7) 12168 = 1+((2)+1+6+8) 12896 = (1+(2))+(8+9+6) 13824 = (1+3+8+(2)+4) 13825 = 1+(3+8+(2)+5) 13828 = 1+3+(8+(2)+8) 13888 = (1+3)+(8+8+8) 15625 = (1+5+6+(2)+5) 15626 = 1+(5+6+(2)+6) 17576 = (1+7+5+7+6) 17577 = 1+(7+5+7+7) 19683 = (1+9+6+8+3) 19684 = 1+(9+6+8+4) 20203 = (2)+0+((2))+((0+3)) 20377 = (2)+0+((3))+(7)+(7) 20536 = ((2))+0+(5)+((3))+(6) 20538 = (2+0+5)+((3))+(8) 20539 = 2+0+(5)+((3))+(9) 20932 = (2)+0+(9)+((3))+((2)) 21411 = ((2+1))+(4+(1+1)) 21475 = ((2+1))+(4)+(7+5) 21743 = ((2)+1)+(7+4)+((3)) 21923 = ((2))+(1+9+2)+((3)) 21944 = ((2+1))+(9+4)+(4) 22394 = 2+((2))+((3))+(9+4) 23276 = 2+((3))+((2)+7)+(6) 23695 = ((2))+((3))+(6+9)+(5) 23781 = 2+((3))+(7+8+1) 23786 = ((2))+((3))+(7+8)+(6) 23788 = 2+((3))+7+(8+8) 23789 = 2+((3))+(7+8)+(9) 25523 = (2)+(5+5+(2))+((3)) 27729 = ((2)+7+7+(2))+(9) 28333 = 2+(8+3)+((3)+3) 29302 = ((2)+9)+((3)+0+2) 30477 = ((3)+0+4)+(7)+(7) 32389 = ((3)+2)+(3+8+9) 32768 = (3+(2)+7+6+8) 33280 = ((3)+3+2)+(8)+0 33281 = ((3)+3+2)+(8)+1 33282 = ((3)+3+2)+(8)+2 33283 = ((3)+3+2)+(8)+3 33284 = ((3)+3+2)+(8)+4 33285 = ((3)+3+2)+(8)+5 33286 = ((3)+3+2)+(8)+6 33287 = ((3)+3+2)+(8)+7 33288 = ((3)+3+2)+(8)+8 33289 = ((3)+3+2)+(8)+9 35512 = ((3)+5)+(5+1+(2)) 36162 = ((3)+6)+1+(6)+(2) 36297 = ((3)+6)+(2)+9+(7) 36307 = (3)+(6+(3))+0+(7) 36344 = ((3)+6)+(3+4)+(4) 36347 = 3+(6+(3))+(4)+(7) 36453 = ((3)+6)+4+(5+3) 36458 = ((3)+6)+4+5+(8) 36582 = ((3)+6)+(5)+(8)+(2) 36672 = ((3)+6)+6+(7+2) 36679 = ((3)+6)+6+7+(9) 36946 = ((3)+6)+9+(4+6) 39316 = 3+9+((3)+1+6) 39392 = ((3))+9+((3))+9+(2) 39443 = ((3))+9+(4)+4+((3)) 43218 = (4+3)+((2+1)+8) 47232 = (4)+(7+2+(3))+((2)) 47239 = (4)+7+((2))+((3)+9) 48392 = (4+8)+((3)+9)+(2) 53396 = (5+(3))+((3))+(9)+(6) 59332 = 5+(9+(3)+3)+(2) 63293 = 6+((3)+(2))+(9)+((3)) 68935 = 6+8+(9+(3)+5) 93312 = (9+(3))+((3)+1+(2)) 93339 = (9+(3))+(3)+((3)+9)

 264 = 2+6+(4) 1386 = 1+(3)+8+(6) 1634 = 1+(6)+(3)+(4) 2401 = (2+4+0+1) 2403 = 2+(4+0+3) 2407 = 2+4+0+(7) 3786 = (3)+(7)+8+(6) 6572 = 6+5+(7+2) 6579 = 6+5+7+(9) 8208 = (8)+(2)+0+(8) 8970 = 8+(9)+(7)+0 8971 = 8+(9)+(7)+1 8972 = 8+(9)+(7)+2 8973 = 8+(9)+(7)+3 8974 = 8+(9)+(7)+4 8975 = 8+(9)+(7)+5 8976 = 8+(9)+(7)+6 8977 = 8+(9)+(7)+7 8978 = 8+(9)+(7)+8 8979 = 8+(9)+(7)+9 9474 = (9)+(4)+(7)+(4) 15283 = 1+(5)+(2)+(8+3) 28570 = 2+(8+5)+7+0 28571 = 2+(8+5)+7+1 28572 = 2+(8+5)+7+2 28573 = 2+(8+5)+7+3 28574 = 2+(8+5)+7+4 28575 = 2+(8+5)+7+5 28576 = 2+8+5+(7+6) 28577 = 2+(8+5)+7+7 28578 = 2+(8+5)+7+8 28579 = 2+(8+5)+7+9 36499 = (3)+(6)+(4+9)+(9) 38564 = 3+(8+5)+(6+4) 65540 = (6+5+5)+4+0 65541 = (6+5+5)+4+1 65542 = (6+5+5)+4+2 65543 = (6+5+5)+4+3 65544 = (6+5+5)+4+4 65545 = (6+5+5)+4+5 65546 = (6+5+5)+4+6 65547 = (6+5+5)+4+7 65548 = (6+5+5)+4+8 65549 = (6+5+5)+4+9 66913 = (6)+(6+9+1)+(3) 67943 = 6+(7+9)+(4+3) 67947 = 6+(7+9)+4+(7) 72097 = (7+2)+(0+9+7) 77937 = (7)+(7+9)+(3+7) 80289 = (8)+((0+2))+(8)+(9) 83728 = (8)+(3+7)+((2))+(8) 94097 = (9+4)+0+(9+7)

 283 = 25+8+35 1274 = (1+2)5+7+45 4150 = 45+1+55+0 4151 = 45+1+55+1 4152 = 45+1+55+2 4153 = 45+1+55+3 4154 = 45+1+55+4 4155 = 45+1+55+5 4156 = 45+1+55+6 4157 = 45+1+55+7 4158 = 45+1+55+8 4159 = 45+1+55+9 16852 = (1+6)+8+5+(2) 17840 = 1+(7)+8+(4+0) 17841 = 1+(7)+8+(4)+1 17842 = 1+(7)+8+(4)+2 17843 = 1+(7)+8+(4)+3 17844 = 1+(7)+8+(4)+4 17845 = 1+(7)+8+(4)+5 17846 = 1+(7)+8+(4)+6 17847 = 1+(7)+8+(4)+7 17848 = 1+(7)+8+(4)+8 17849 = 1+(7)+8+(4)+9 20175 = (2+0+1)+(7)+(5) 23057 = (2+3)+(0+5)+(7) 24597 = (2+4)+5+9+(7) 32780 = 3+2+7+(8+0) 32781 = 3+2+7+(8)+1 32782 = 3+2+7+(8)+2 32783 = 3+2+7+(8)+3 32784 = 3+2+7+(8)+4 32785 = 3+2+7+(8)+5 32786 = 3+2+7+(8)+6 32787 = 3+2+7+(8)+7 32788 = 3+2+7+(8)+8 32789 = 3+2+7+(8)+9 34035 = (3)+(4)+(0+3+5) 34038 = 3+(4)+(0+3)+(8) 36385 = (3)+6+(3)+(8)+(5) 54748 = (5)+(4)+(7)+(4)+(8) 55825 = (5)+(5)+(8)+(2+5) 55827 = (5)+(5)+(8)+2+(7) 57356 = 5+(7)+(3+5)+(6) 59060 = 5+(9)+0+6+0 59061 = 5+(9)+0+6+1 59062 = 5+(9)+0+6+2 59063 = 5+(9)+0+6+3 59064 = 5+(9)+0+6+4 59065 = 5+(9)+0+6+5 59066 = 5+(9)+0+6+6 59067 = 5+(9)+0+6+7 59068 = 5+(9)+0+6+8 59069 = 5+(9)+0+6+9 59332 = 5+(9)+(3)+3+(2) 69965 = 6+(9)+9+(6)+(5) 76894 = (7)+6+8+(9)+(4) 82378 = (8)+(2)+3+(7)+(8) 91820 = (9)+1+(8)+2+0 91821 = (9)+1+(8)+2+1 91822 = (9)+1+(8)+2+2 91823 = (9)+1+(8)+2+3 91824 = (9)+1+(8)+2+4 91825 = (9)+1+(8)+2+5 91826 = (9)+1+(8)+2+6 91827 = (9)+1+(8)+2+7 91828 = (9)+1+(8)+2+8 91829 = (9)+1+(8)+2+9 92727 = (9)+(2)+(7)+(2)+(7) 93084 = (9)+(3)+0+(8)+(4) 95797 = 9+(5)+(7)+(9)+(7)

 50764 = 5+0+7+66+46 132 = 1+3+27 262 = 27+6+27 16524 = 1+6+5+27+47 78140 = 7+8+(1+4)7+0 78141 = 7+8+(1+4)7+1 78142 = 7+8+(1+4)7+2 78143 = 7+8+(1+4)7+3 78144 = 7+8+(1+4)7+4 78145 = 7+8+(1+4)7+5 78146 = 7+8+(1+4)7+6 78147 = 7+8+(1+4)7+7 78148 = 7+8+(1+4)7+8 78149 = 7+8+(1+4)7+9 78275 = 7+8+27+7+57 94520 = 9+47+57+2+0 94521 = 9+47+57+2+1 94522 = 9+47+57+2+2 94523 = 9+47+57+2+3 94524 = 9+47+57+2+4 94525 = 9+47+57+2+5 94526 = 9+47+57+2+6 94527 = 9+47+57+2+7 94528 = 9+47+57+2+8 94529 = 9+47+57+2+9 94652 = 9+47+6+57+27 72104 = 7+(2+1)8+0+48 179212 = 1+7+9+(2+1)11+211 179213 = 1+7+9+211+1+311 8200 = 8+213+0+0 8201 = 8+213+0+1 8202 = 8+213+0+2 8203 = 8+213+0+3 8204 = 8+213+0+4 8205 = 8+213+0+5 8206 = 8+213+0+6 8207 = 8+213+0+7 8208 = 8+213+0+8 8209 = 8+213+0+9

Here are the small solutions for multiplication:

Squares with Multiplication
 128 = 1*((2))*8 175 = 1*7*(5) 384 = 3*8*(4) 672 = 6*7*((2)) 735 = (7)*3*5 1296 = 1*(2)*9*(6) 6144 = 6*1*4*((4)) 18432 = 1*(8)*(4)*(3)*2 23328 = 2*((3))*(3)*2*8 34992 = 3*4*9*(9*2) 82944 = 8*2*(9*4)*4 442368 = (4)*(4)*(2)*(3)*6*8 1492992 = 1*((4))*9*(2)*9*9*2 2333772 = 2*(3)*(3)*3*(7)*(7)*2 2612736 = ((2))*(6)*1*2*7*(3)*(6) 3981312 = (3)*9*((8))*1*3*1*(2) 4128768 = (4)*1*((2))*8*7*(6)*8 9289728 = 9*((2))*(8)*9*7*2*8 12192768 = 1*((2))*1*(9)*(2)*(7)*6*8 13226976 = 1*((3))*(2)*2*6*(9)*7*6 13395375 = 1*(3)*(3)*9*(5)*3*(7)*5 13436928 = 1*((3))*4*3*6*9*(2)*(8) 21233664 = (((2)))*1*((2))*3*3*6*6*(4) 22127616 = ((2))*(2)*1*(2)*((7))*6*1*6 27869184 = (2)*7*8*6*(9*1*8)*4 49787136 = (4)*9*(7)*8*(7)*1*3*6 161243136 = 1*((6*1*2))*(4*3*1*3)*6 376233984 = (3)*7*(6)*2*3*3*9*(8*4) 429981696 = (4)*((2))*9*9*(8)*1*6*9*6 672126336 = ((6*7))*2*1*2*6*3*3*6 1269789696 = 1*2*6*9*7*(8*9)*6*9*6 1416167424 = 1*4*1*6*1*6*((7*4))*(2)*4

Cubes with Multiplication
 135 = 1*(3)*5 1715 = 1*(7)*1*5 23328 = 2*(3*3)*2*8 2239488 = 2*2*3*(9)*4*8*8 2333772 = 2*3*3*(3*7)*7*2 9289728 = 9*2*8*9*7*2*(8) 21233664 = ((2))*1*2*3*3*6*6*(4) 27869184 = (2)*7*8*(6)*9*1*8*4

Here are the small solutions for factorials:

 144 = (1+4)!+4! 145 = 1+4!+5! 733 = 7+3!+3!! 5160 = 5!+(1+6)!+0 5161 = 5!+(1+6)!+1 5162 = 5!+(1+6)!+2 5163 = 5!+(1+6)!+3 5164 = 5!+(1+6)!+4 5165 = 5!+(1+6)!+5 5166 = 5!+(1+6)!+6 5167 = 5!+(1+6)!+7 5168 = 5!+(1+6)!+8 5169 = 5!+(1+6)!+9 40440 = (4+(0))+(4+4)+0 40441 = (4+(0))+(4+4)+1 40442 = (4+(0))+(4+4)+2 40443 = (4+(0))+(4+4)+3 40444 = (4+(0))+(4+4)+4 40445 = (4+(0))+(4+4)+5 40446 = (4+(0))+(4+4)+6 40447 = (4+(0))+(4+4)+7 40448 = (4+(0))+(4+4)+8 40449 = (4+(0))+(4+4)+9 40584 = (4+0!)!+5!+8!+4! 40585 = 4!+0!+5!+8!+5!

Factorials with Multiplcation
 36 = 3!*6 82944 = 8*2*9*4!*4!

Here are the small solutions for other bases:

Base 2
 11001 = ((1+1)+0+0+1) 1010001 = ((1+0+1+0+0+0+1)) 1111001 = ((1+1+1)+1+0+0+1) 100100001 = (((1+0+0+1))+0+0+0+0+1) 101101001 = (((1+0+1))+1+0+1+0+0+1) 110111010 = (((1+1))+(0+1+1)+1)+0+1+0 110111011 = (((1+1))+(0+1+1)+1)+0+1+1

Base 3
 11 = (1+1) 12 = 1+(2) 22 = (2)+(2) 121 = (1+2+1) 122 = 1+(2+2) 202 = ((2))+0+(2) 221 = (2+2+1) 1012 = ((1+0+1))+((2)) 1112 = ((1+1))+(1+(2)) 1211 = (1+2+(1+1)) 1212 = 1+(2+1+(2)) 1222 = (1+2+(2))+(2) 2101 = ((2)+(1+0+1)) 2112 = (2)+((1+1)+(2)) 2222 = (2+2)+((2)+(2)) 10121 = ((1+0+1))+((2+1)) 10122 = 1+0+((1+2))+((2)) 10201 = (1+0+(2+0+1)) 10212 = ((1+0+2)+1)+(2) 10221 = ((1+0+2))+((2)+1) 11022 = (1+(1+0+2))+((2)) 11111 = ((1+1+1)+1+1) 11112 = 1+((1+1+1)+2) 11122 = ((1+1+1)+2)+(2) 12222 = ((1+2))+((2))+((2)+(2)) 20021 = ((2)+(0+0+2+1)) 20212 = ((2))+((0+2)+(1+2)) 21021 = ((2)+(1+0+2)+1) 21212 = ((2))+((1+2)+1+(2)) 22122 = ((2)+(2))+((1+2)+(2)) 22221 = ((2))+((2)+2+(2+1))

Base 4
 21 = (2+1) 23 = 2+(3) 221 = ((2))+((2)+1) 1101 = ((1+1+0+1)) 1102 = 1+((1+0+2)) 1103 = 1+1+0+((3)) 1123 = 1+((1+2))+(3) 1210 = (1+(2+1))+0 1211 = ((1+2)+1)+1 1212 = ((1+2)+1)+2 1213 = ((1+2)+1)+3 1231 = (1+2)+((3)+1) 1232 = (1+(2))+((3))+(2) 1322 = 1+(3+(2)+(2)) 2221 = ((2)+(2)+(2)+1) 2223 = 2+(2+2+(3)) 1313 = 2+((3))+(1+(3)) 3303 = ((3))+((3))+0+((3)) 3322 = ((3))+((3)+2+2)

Base 5
 23 = (2)+(3) 31 = (3+1) 33 = (3)+(3) 34 = 3+(4) 112 = ((1+1))+((2)) 121 = (1+(2)+1) 122 = 1+(2+(2)) 224 = (2+2+4) 243 = ((2)+4)+(3) 320 = ((3))+(2)+0 321 = ((3))+(2)+1 322 = ((3))+(2)+2 323 = ((3))+(2)+3 324 = ((3))+(2)+4 330 = (3)+((3))+0 331 = (3)+((3))+1 332 = (3)+((3))+2 333 = (3)+((3))+3 334 = (3)+((3))+4 423 = (4)+((2))+((3)) 431 = (4)+((3)+1)

Base 6
 41 = (4+1) 45 = 4+(5) 221 = (2)+((2+1)) 223 = 2+(2)+((3)) 225 = ((2)+(2))+(5) 243 = 2+(4)+((3)) 244 = (2+4+4) 312 = ((3)+1)+((2)) 334 = ((3))+(3+4) 532 = (5+(3))+(2)

Base 7
 12 = (1+2) 13 = 1+(3) 22 = (2+2) 24 = 2+(4) 34 = (3)+(4) 44 = (4)+(4) 51 = (5+1) 56 = 5+(6) 63 = (6)+(3) 122 = 1+((2)+(2)) 144 = (1+4+4) 145 = 1+(4+5) 153 = 1+5+((3)) 232 = ((2)+3+(2)) 264 = (2+6+4) 266 = 2+(6+6) 321 = ((3))+((2+1)) 323 = ((3))+2+((3)) 331 = ((3)+3+1) 333 = ((3))+((3))+(3) 334 = 3+((3)+4) 343 = ((3))+(4)+((3)) 346 = ((3))+(4+6) 463 = (4)+(6+(3)) 522 = 5+((2+2)) 524 = 5+2+((4)) 534 = 5+(3)+((4)) 544 = 5+((4))+(4) 614 = (6+1)+((4)) 646 = (6)+((4))+(6)

Base 8
 20 = ((2))+0 21 = ((2))+1 22 = ((2))+2 23 = ((2))+3 24 = ((2))+4 25 = ((2))+5 26 = ((2))+6 27 = ((2))+7 61= (6+1) 64 = (6)+(4) 67 = 6+(7) 133 = 1+(3)+((3)) 153 = 1+(5)+((3)) 172 = 1+(7+(2)) 400 = ((4))+0+0 401 = ((4))+0+1 402 = ((4))+0+2 403 = ((4))+0+3 404 = ((4))+0+4 405 = ((4))+0+5 406 = ((4))+0+6 407 = ((4))+0+7 420 = ((4))+((2))+0 421 = ((4))+((2))+1 422 = ((4))+((2))+2 423 = ((4))+((2))+3 424 = ((4))+((2))+4 425 = ((4))+((2))+5 426 = ((4))+((2))+6 427 = ((4))+((2))+7 461 = ((4))+(6+1) 464 = ((4))+(6)+(4) 467 = ((4))+6+(7) 752 = (7)+(5+((2)))

Base 9
 22 = (2+(2)) 45 = (4)+(5) 55 = (5)+(5) 71 = (7+1) 72 = (7)+((2)) 78 = 7+(8) 121 = (1+(2+1)) 235 = ((2)+(3))+(5) 237 = ((2)+3+7) 287 = ((2))+(8+7) 322 = 3+(((2)))+(2) 334 = (3)+(3)+((4)) 342 = (3)+((4))+((2)) 345 = 3+((4))+(5) 352 = (3)+(5)+(((2))) 364 = (3)+(6)+((4)) 372 = 3+(7)+(((2))) 432 = ((4))+((3))+((2)) 484 = ((4))+(8+4) 522 = (5)+(((2))+(2)) 540 = (5+(4))+0 541 = (5+(4))+1 542 = (5+(4))+2 543 = (5+(4))+3 544 = (5+(4))+4 545 = (5+(4))+5 546 = (5+(4))+6 547= (5+(4))+7 548 = (5+(4))+8 625 = (6+((2)))+(5) 728 = (7+((2)))+(8)

Mark Michell suggested that some numbers are non-trivially S-numbers using non-integer powers. One example: 8107 = (2+2)P + (3+5)P, where P=log29.

Bryce Herdt suggested that we consider squaring and subtracting. His solutions are shown below.

Squares with Subtraction
 –5904 = (–5)–((9)–0–4) –4112 = –(((4–1)–1))–((2)) –2394 = ((–2))–(3)–(9–(4)) –2337 = (–2–3–3)–((7)) –512 = –((5–1))–(((2))) –264 = –2–6–((4)) –192 = ((–1)–9)–(((2))) –89 = –8–(9) –48 = (–4)-(8) 49 = ((4)–9) 196 = (1–9–6) 207 = (((2)))–0–(7) 224 = (((2)))–((2))–(4) 240 = (((2)))–(4)–0 247 = (((2)))–(4–7) 248 = (((2–4)))–8 572 = ((5)–(7))–(2) 573 = ((5)–(7))–3 1296 = (((1–(2))–9–6)) 1762 = (1–7–(6))–2 2391 = ((2–(3)))–9–1 2397 = ((2–(3)))–(9–7) 2401 = ((2–(4–0–1))) 3969 = ((3–9–6)–(9)) 4088 = (((4)–0–8))–8 4624 = (((4)–6)–((2))–(4)) 5472 = (((5)–(4))–7)–(2) 5473 = (((5)–(4))–7)–3 8462 = (8–((4)–6))–2

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 8/24/11.