Funny Arithmetic

The following operators are allowed:

• arithmetic
  • +, addition
  • -, subtraction
  • ·, multiplication
  • /, division
  • %, modulus
• logical (bitwise)
  • &, conjunction

    &	0	1
    0	0	0
    1	0	1

  • |, disjunction

    |	0	1
    0	0	1
    1	1	1

  • ^, exclusive or

    ^	0	1
    0	0	1
    1	1	0

• bitwise shift
  • <<, left shift. For example,
    3<<2=12, 5<<1=10
  • >>, right shift. For example,
    5>>1=2, 13>>2=3

From time to time I am asked questions of number representations related to the above. I'll be orginizing results in tables to keep them easily available. Normally, in this kind of problems, allowed operations are different. I was limited by the JavaScript built-in parsing facility. Thus, in the following pages I shall use operations listed below:

Remark: the whole part function is a powerful addition to the common arithmetic operations and factorials. This can be demonstrated by the surprising, but shallow, identity: [π]^[e] + [e] = [e]^[π] + [π].

Everyone can take part in filling these tables or creating new ones. I insist on one condition though:

• One 4
  • One 4, a story
• Three 3's
• Three 4's
• Three 5's
• Four 3's
• Four 4's
• Four 5's
• Fun With Digits
• Any integer expressed with digits 0-9 each used only once
• A 9's Fan's Clock

Following is a list of simple facts that frequently proves handy solving this kind of puzzles: for every positive number a:

1. a - a = 0
2. a / a = 1
3. (a + a) / a = 2
4. (a + a + a) / a = 3
5. √a·√a = a

For a digit n there are additional useful identities:

1. n / .(n) = 9
2. n / .n = 10
3. nn / n = 11

(A Java applet that could help experiment with this kind of problems is also avaialble.)