Can You Find The FUN?

Riddler Express

From Dean Ballard comes something quite “fun”:

If you use each of the counting numbers 1, 2 and 3 at most once, and you can add or multiply them in any order you choose, you can make all the numbers from 1 to 9:

1. 1
2. 2
3. 3
4. 1+3
5. 2+3
6. 2×3
7. 2×3+1
8. (3+1)×2
9. (2+1)×3

However, there is no way to make the number 10. Therefore, 10 is the First Unmakeable Number, or “FUN,” for the first three counting numbers.

What is the FUN for the first four counting numbers?

Extra credit: How many other “FUN numbers” can you find?

Solution

1. 1
2. 2
3. 3
4. 4
5. 1+4
6. 2+4
7. 3+4
8. 1+3+4
9. 2+3+4
10. 1+2+3+4
11. 2*4+3
12. 3*4
13. 3*4+1
14. 3*4+2
15. (1+4)*3
16. (1+3)*4

Answer

1-16 were straightforward. 17 seems to be FUN. Since it is prime, as a check, I used the following four questions:

• Can I achieve 13 without a 4?
• Can I achieve 14 without a 3?
• Can I achieve 15 without a 2?
• Can I achieve 16 without a 1?

Extra Credit

For the first five counting numbers:

• ...
• 53 = 1+(2*5+3)*4
• 54 = (2*3)*(4+5)
• 55 = (2*4+3)*5
• 56 = (1+2+4)*(3+5)
• 57 = ((1+2)*5+4)*3

Using a similar line of questioning from above, as well as the impossibility of achieving 29x2, 58 seems to be FUN.

Rohan Lewis

2022.06.05