January 2013
Problem of the Month

The New Year Challenge

Every year from 1968 until 1999, I posed the following challenge to my students:

Write expressions for all the numbers from 1 to 100 using only the digits in the current year in order and using the operations +, -, x, ÷ (or / for divided by), ^ (raised to a power), sqrt (square root), ! (factorial), and int (or [ ] for greatest integer function) along with grouping symbols.

So, the first problem of the new year is to use only the digits 2, 0, 1, 3, (and in that order) along with the operations listed above to write expressions for all the numbers from 0 to 21.   For example, the number 67 could be expressed as 201 / 3.

Extra credit for those who can go past 21 (consecutively).

Some Solutions to the Problem:

0 = 2 * 0 * 13   or   2 + 0 + 1 - 3
  or   2*0*1*3
1 = -2 + 0 * 1 + 3   or   (2+0+1)/3
2 = 2 + 0 * 13   or   2+0*1*3
3 = 2 + 0 + 1^3   or   2*0*1+3
4 = -2 + 0 + 1 + 3   or   2+0-1+3
  or   2*0+1+3
5 = 2 + 0 + 1 * 3   or   2+0*1+3
6 = 2 + 0 + 1 + 3   or   2*0*1+(3!)
7 = 20 - 13   or   (20+1)/3
8 = 2 + 0 * 1 + 3!   or   (2+0)*(1+3)
9 = - int(sqrt(20)) + 13   or   (2 + 0 + 1) * 3
  or   2 + 0 + 1 + 3!   or   (2+0+1)*3
10 = 20 / (-1 + 3)
  or   INT(SQRT(20))^1+(3!)
11 = -2 + 0 + 13   or   int(sqrt(201))-3
  or   INT(SQRT(20))+1+(3!)
12 = -(2^0) + 13   or   (2+0+1)!+3!
  or   INT(SQRT(20))*1*3
13 = 2 * 0 + 13   or   20-1-3!
14 = 20 - 1 * 3!   or   2^0 + 13
  or   20/1-3!   or   INT(SQRT(2)+0+13)
15 = 20 + 1 - 3!   or   2+0+13
16 = 20 - 1 - 3
17 = 20 * 1 - 3   or   20/1-3
18 = 20 + 1- 3
19 = 20 - (1^3)
20 = 20 * (1^3)   or   int(sqrt(201))+3!
21 = 20 + 1^3
22 = 20 - 1 + 3
23 = 20 * 1 + 3   or   20/1+3   or   20+1*3
24 = 20 + 1 + 3   or   2*0+(1+3)!
25 = 20 - 1 + 3!   or   20-1+(3!)
26 = 20 * 1 + 3!   or   20/1+3!
  or   2+0+(1+3)!
27 = 20 + 1 + 3!   or   (2+0+1)^3
28 = 2 * (0! + 13)
  or   INT(SQRT(20))+(1+3)!
  or   int(sqrt(201))*int(sqrt(3!))
29 = int(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(201!)))))))))*int(sqrt(3))
30 = (2+0!+1)!+3!
31 = int(sqrt(sqrt(sqrt(sqrt(24!))))*sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(3)))))))
32 = int(sqrt(sqrt(sqrt(sqrt(24!))))*sqrt(sqrt(sqrt(sqrt(3)))))
33 = int(sqrt(sqrt(sqrt(sqrt(24!)))))*int(sqrt(sqrt(sqrt(sqrt(3)))))
34 = int(sqrt(sqrt(sqrt(sqrt(24!*3!)))))
35 = int(sqrt(sqrt(sqrt(sqrt(24!))))*sqrt(sqrt(sqrt(3))))


James Alarie gets extra credit for solving 1 through 35.
Mike Bova gets extra credit for solving 1 through 28.

Correctly solved by:

1. James Alarie	Flint, Michigan
2. Mike Bova	Wallingford Connecticut

Send any comments or questions to: David Pleacher