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Allowed Operations ...Addition, Subtraction, Multiplication, Division, Exponentiation, Simple Factorial. Left and Right Brackets allowed.

Expression should involve minimum number of total characters:

Example:

Number 7. Square. 49. Sum of Digits. 13

Using only digits 1 and 3, express 49

One possibility is. 49 = (3! + 1) * (3! + 1)

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  • $\begingroup$ Sure..only restriction is among all the possibilities with allowed signs, come up with minimum footprint $\endgroup$
    – Uvc
    Commented May 30, 2019 at 15:30
  • $\begingroup$ is square root allowed? $\endgroup$
    – Omega Krypton
    Commented May 30, 2019 at 15:30
  • $\begingroup$ No..only given signs(operations allowed) $\endgroup$
    – Uvc
    Commented May 30, 2019 at 15:31
  • $\begingroup$ Are concatenations allowed? E.g. given 13 in your example, could we use the number 13, or only 1 and 3? $\endgroup$
    – Rand al'Thor
    Commented May 30, 2019 at 15:33
  • $\begingroup$ No concatenation..only mentioned operations..in the example, 1 and 3 can be used with any mentioned allowable operations $\endgroup$
    – Uvc
    Commented May 30, 2019 at 15:35

3 Answers 3

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1: sod=1

$1$

4: sod=4

$4$

9: sod=9

$9$

16: sod=7

$(7+\frac77)(\frac{7+7}{7})$

25: sod=7

$(7-\frac{7+7}{7})^{\frac{7+7}{7}}$

36: sod=9

$9+9+9+9$

49: sod=13

$(3!+1)^{1+1}$

64: sod=10

$(1+1)^{(1+1+1)!}$

81: sod=9

$9\times9$

100: sod=1

$((1+1+1)^{1+1}+1))^{1+1}$

121: sod=4

$(4+\frac44)!+\frac44$

144: sod=9

$(9-\frac99)(9+9)$

169: sod=16

$(6+6+1)^{1+1}$

196: sod=16

$(6+6+1+1)^{1+1}$

225: sod=9

$9\times(9+9+9)-9-9$

256: sod=13

$(1+1)^{(1+1)^3}$

289: sod=19

$(9+9-1)^{1+1}$

324: sod=9

$(9+9)(9+9)$

361: sod=10

$\frac{(1+1+1)!!}{1+1}+1$

400: sod=4

$4\times(4\cdot4!+4)$

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  • $\begingroup$ no- the 10 is not allowed... sorry $\endgroup$
    – Omega Krypton
    Commented May 30, 2019 at 15:39
  • $\begingroup$ No concatenation of 1 and 0 allowed $\endgroup$
    – Uvc
    Commented May 30, 2019 at 15:41
  • $\begingroup$ why does the 400 case work? 4.4! is like 48 point something $\endgroup$
    – Omega Krypton
    Commented May 30, 2019 at 15:47
  • $\begingroup$ using dot notation for multiply @OmegaKrypton $\endgroup$
    – JMP
    Commented May 30, 2019 at 15:48
  • $\begingroup$ good game! +1 JMP! $\endgroup$
    – Omega Krypton
    Commented May 30, 2019 at 15:49
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Partial

notation: sod = sum of digits, sq = square

1) sq=1, sod=1

$1=1$

2) sq=4, sod=4

$4=4$

3) sq=9, sod=9

$9=9$

4) sq=16, sod=7

$16=7+7+7/7+7/7$

5) sq=25, sod=7

6) sq=36, sod=9

$36=9+9+9+9$

7) sq=49, sod=13

$49=(3!+1)*(3!+1)$ (from OP)

8) sq=64, sod=10

$64=\big((1+1)^{(1+1+1)}\big)^{(1+1)}$

17) sq=289, sod=19

289=(9+9-1)^(1+1)

18) sq=324, sod=9

324=(9+9)^(9/9+9/9)

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  • $\begingroup$ @omega..answers for 19 and 20 missing $\endgroup$
    – Uvc
    Commented May 30, 2019 at 22:38
  • $\begingroup$ @Uvc sorry, didnt bother to finish this while Rand and JMP have finished... ;) $\endgroup$
    – Omega Krypton
    Commented May 31, 2019 at 0:07
  • $\begingroup$ Still..whatever you have done is pretty good. $\endgroup$
    – Uvc
    Commented May 31, 2019 at 0:08
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  1. Square 1. Sum of digits 1.

    $1=1$.

  2. Square 4. Sum of digits 4.

    $4=4$.

  3. Square 9. Sum of digits 9.

    $9=9$.

  4. Square 16. Sum of digits 7.

    $16=7+7+\frac{7}{7}+\frac{7}{7}$.

  5. Square 25. Sum of digits 7.

    $25=\frac{(7\times7)+\frac{7}{7}}{\frac{7}{7}+\frac{7}{7}}$.

  6. Square 36. Sum of digits 9.

    $36=9+9+9+9$.

  7. Square 49. Sum of digits 13.

    $49=(3!+1)\times(3!+1)$.

  8. Square 64. Sum of digits 10.

    $64=(1+1+1+1)^{1+1+1}$.

  9. Square 81. Sum of digits 9.

    $81=9\times9$.

  10. Square 100. Sum of digits 1.

    $100=((1+1+1+1+1)\times(1+1))^{1+1}$.

  11. Square 121. Sum of digits 4.

    $121=(4+4+4-\frac{4}{4})^{\frac{4}{4}+\frac{4}{4}}$.

  12. Square 144. Sum of digits 9.

    $144=9\times(9+9)-9-9$.

  13. Square 169. Sum of digits 16. (Most interesting one!)

    $169=(6-1)!+(6+1)^{1+1}$.

  14. Square 196. Sum of digits 16.

    $196=(6+6+1+1)^{1+1}$.

  15. Square 225. Sum of digits 9.

    $225=9\times(9+9+9)-9-9$.

  16. Square 256. Sum of digits 13.

    $256=(3+1)^{3+1}$.

  17. Square 289. Sum of digits 19.

    $289=(9+9-1)^{1+1}$.

  18. Square 324. Sum of digits 9.

    $324=(9+9)\times(9+9)$.

  19. Square 361. Sum of digits 10.

    $361=\frac{(1+1+1+1+1+1)!}{1+1}+1$.

  20. Square 400. Sum of digits 4.

    $400=4\times(4\times4!+4)$.

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1
  • $\begingroup$ good game! +1 Rand! $\endgroup$
    – Omega Krypton
    Commented May 30, 2019 at 15:49

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