# Smallest number to be made in one way

Take the numbers 1,3,4,6,9,10. We can make other numbers using these numbers and the operations +,-,*,/. E.g. we can make 244 with:

10 * 3 = 30
30 - 6 = 24
9 + 1 = 10
24 * 10 = 240
240 + 4 = 244


In this case there is more than one way to do it. For example,

3 + 1 = 4
10 * 6 * 4 = 240
240 + 4 = 244


This second solution also doesn't use all the given numbers.

There is only one way to make 713, that is:

9 * 6 = 54
54 - 3 = 51
10 + 4 = 14
51 * 14 = 714
714 - 1 = 713


What is the smallest positive integer that can be made in exactly one way and also requires using each one of 1,3,4,6,9,10? You can use each number exactly once.

As an example, 10 is no good as a target number as you can make it just using the number 10.

• @bobble you can use a computer if you prefer.
– Simd
Commented Jan 21, 2023 at 4:21
• What is a different way ? Is 1/(3/6) differnt from 6/3/1 ? Commented Jan 24, 2023 at 8:16
• Has a correct answer been given? If so, please don't forget to $\color{green}{\checkmark \small\text{Accept}}$ it :)
– Rubio
Commented Jan 27, 2023 at 11:39
• @Rubio it hasn't been given yet
– Simd
Commented Jan 27, 2023 at 11:48

the answer that I found is

695

with

$$9+3 = 12$$

$$12*10 = 120$$

$$120-4 = 116$$

$$116*6=696$$

$$696-1=695$$

I wrote a program and all other numbers below 695 has multiple answers.

• 4 + 3 = 7 10 * 9 * 7 = 630 630 - 1 = 629 629 + 6 = 635 9 * 6 = 54 54 + 4 = 58 10 + 1 = 11 58 * 11 = 638 638 - 3 = 635
– Simd
Commented Jan 21, 2023 at 19:44
• @Simd editted again :)
– Oray
Commented Jan 21, 2023 at 21:42
– Simd
Commented Jan 27, 2023 at 11:47

Here's what I've come up with. I'll keep playing around ;^)

4 * 3 = 12
12 / 6 = 2
2 * 10 = 20
20 - 9 = 11
11 - 1 = 10

Incidentally, is there a definitive correct answer? I.e. one you generated with an alogrithm beforehand.

• Another way is $(10+9+1+6+4)/3$ Commented Jan 20, 2023 at 23:02
• Good one - I'll admit a I'm a bit fuzzy on the rules, was going by OP's lead. Commented Jan 20, 2023 at 23:04
• You can make 10 just using the number 10 as well so this isn’t allowed. I added a clarification.
– Simd
Commented Jan 21, 2023 at 4:20
• @Simd but using just the number 10 isn't using all numbers exactly once? So for the purpose of the condition "exactly one way", we are also including those which don't use all numbers? Commented Jan 21, 2023 at 5:12
• @justhalf Yes. I will try to make it clearer in the question.
– Simd
Commented Jan 21, 2023 at 6:16