Question

6) Use any of the digits 1, 3, and 9 and the operation signs +, -, x, to write all the whole numbers from 1 through 13. Each digit can be expressed only once in each example. You can use other digits in the expression, but you must also use a 1, 3, or 9 at least once in each expression.

Example: The first three (3) have some examples for you.

Number


a) 1

Expression


2 - 1 OR 3 - 2

b) 2

3 - 1

c) 3

3 x 1 OR 9 3

d) 4

e) 5

f) 6

g) 7

h) 8

i) 9

j) 10

k) 11

l) 12

m) 13

Answer 1

1 = 1; 2 = 3 -1; 3 = 3; 4 = 3 +1; 5 = 9 -3 -1;

6 = 9 -3; 7 = 9 -3 +1; 8 = 9 -1; 9 = 9; 10 = 9 +1

11 = 9 +3 -1; 12 = 9 +3; 13 = 9 +3 +1

Explanation:

You want the numbers 1 – 13 expressed in terms of the digits 1, 3, 9 using operations +, -, and ×.

Base 3

The digits 1, 3, 9 represent the place values of numbers in base 3. This means we can use the base-3 representation of a number to give a clue as to how to represent it using these digits.

The digits of a base 3 number are 0, 1, 2. We don't have a 2 to work with, but we know that 2 = 3 -1, so we can use that fact. Here is an example:

5 = 12₃ = 1×3 + (3 -1)×1 = 3 +3 -1

= 20₃ -1 = (3 -1)×3 -1 = 9 -3 -1

After writing a few numbers, we notice the signs go in the progression +, -, 0 where 0 means the digit is not included. The attachment shows the sums that make the numbers 1–13.

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Additional comment

We could, of course, use the allowed "other digits" to include 2. For example, ...

5 = 3 + 2×1

6 = 2×3

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