# Express the Squares of first 20 Numbers with only numbers found in their Sum of Digits

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)

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

## 3 Answers

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

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

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

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)

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

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

• good game! +1 Rand! Commented May 30, 2019 at 15:49