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# Missing 9s

**Answer**: 363

**Counting 300 places on Sara's list**

Between 1 and 100, Sara won't list 9, 19, 29, 39, 49, 59, 69, 79, 89 or 90 - 99.

19 numbers will be missed between 1 and 100, 81 will be listed

100 will be the 81st number

81 numbers will be listed between 101 and 200

200 will be the 162nd number

81 numbers will be listed between 201 and 300

300 will be the 243rd number

301 will be the 244th number

...

308 will be the 251st number

310 will be the 252nd number

...

318 will be the 260th number

320 will be the 261st number

...

328 will be the 269th number

330 will be the 270th number

...

340 will be the 279th number

...

350 will be the 288th number

...

360 will be the 297th number

...

363 will be the 300th number

**Counting in blocks with a diagram**

Representing the numbers 1 to 100 in a square, the numbers Sara won't list are shown in red.

The numbers up to 100 on Sara's list make a 9 by 9 square.

81 numbers in this square

The square from 101 to 200 is similar

81 numbers in this square makes 162 altogether

The square from 201 to 300 is similar

81 numbers in this square makes 243 altogether

57 more numbers to reach 300

Rows from the square from 301 to 400

The rows are all 9 numbers long

9$\times$6 = 54 numbers in this block

3 more numbers to reach 300

300th number is 363

**Using base 9**

Numbers written in base 9 do not contain any 9s.

In base 9, the numbers 1, 2, 3, 4, 5, 6, 7, 8 are written as usual, but the number '9' is written as 10. Instead of having a ones column and a tens column and a hundreds (=10$^2$) column and so on, base 9 has a ones column and a nines column and an eighty-ones (=9$^2$) column and so on. For example, '18' is written as 20 because it is 2 nines and 0 ones, and '82' is written as 101 because it is 1 eighty-one, 0 nines and 1 one.

To find the 300th number on Sara's list, write 300 in base 9.

3$\times$81 = 243 < 300

4$\times$81 = 324 > 300

There are 3 eighty-ones in 300, remainder 57.

57 = 6$\times$9 + 3,

So 300 = 3$\times$81 + 6$\times$9 + 3

In base 9 it is written 363.

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Age 11 to 14

ShortChallenge Level

- Problem
- Solutions

Between 1 and 100, Sara won't list 9, 19, 29, 39, 49, 59, 69, 79, 89 or 90 - 99.

19 numbers will be missed between 1 and 100, 81 will be listed

100 will be the 81st number

81 numbers will be listed between 101 and 200

200 will be the 162nd number

81 numbers will be listed between 201 and 300

300 will be the 243rd number

301 will be the 244th number

...

308 will be the 251st number

310 will be the 252nd number

...

318 will be the 260th number

320 will be the 261st number

...

328 will be the 269th number

330 will be the 270th number

...

340 will be the 279th number

...

350 will be the 288th number

...

360 will be the 297th number

...

363 will be the 300th number

Representing the numbers 1 to 100 in a square, the numbers Sara won't list are shown in red.

The numbers up to 100 on Sara's list make a 9 by 9 square.

81 numbers in this square

The square from 101 to 200 is similar

81 numbers in this square makes 162 altogether

The square from 201 to 300 is similar

81 numbers in this square makes 243 altogether

57 more numbers to reach 300

Rows from the square from 301 to 400

The rows are all 9 numbers long

9$\times$6 = 54 numbers in this block

3 more numbers to reach 300

300th number is 363

Numbers written in base 9 do not contain any 9s.

In base 9, the numbers 1, 2, 3, 4, 5, 6, 7, 8 are written as usual, but the number '9' is written as 10. Instead of having a ones column and a tens column and a hundreds (=10$^2$) column and so on, base 9 has a ones column and a nines column and an eighty-ones (=9$^2$) column and so on. For example, '18' is written as 20 because it is 2 nines and 0 ones, and '82' is written as 101 because it is 1 eighty-one, 0 nines and 1 one.

To find the 300th number on Sara's list, write 300 in base 9.

3$\times$81 = 243 < 300

4$\times$81 = 324 > 300

There are 3 eighty-ones in 300, remainder 57.

57 = 6$\times$9 + 3,

So 300 = 3$\times$81 + 6$\times$9 + 3

In base 9 it is written 363.

You can find more short problems, arranged by curriculum topic, in our short problems collection.