Funny Arithmetic
The following operators are allowed:
- arithmetic
- +, addition
- -, subtraction
- ·, multiplication
- /, division
- %, modulus
- logical (bitwise)
- bitwise shift
- <<, left shift. For example,
3<<2=12, 5<<1=10 - >>, right shift. For example,
5>>1=2, 13>>2=3
- <<, left shift. For example,
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:
- +(addition), -(subtraction), ·(multiplication), /(division)
- !(factorial, 5!=5·4·3·2·1, 6!=6·5!)
- !!(double factorial, 5!!=5·3·1, 6!!=6·4·2)
- !(subfactorial, !n = n!·(1/2! - 1/3! + ... + (-1)^{n}/n!), e.g,
!4 = 24(1/2! - 1/3! + 1/4!) = 9. ) - a^{b}(power: a to the power of b), √(square root)
- .(decimal point: 4/.4=10)
- .(periodic fraction: =4/.444...=9. For simplicity sake I'll write, say, 4/.(4) or with an overline, like, 4/.4)
- [] (whole part: [1.7]=1, [3.6]=3)
- any kind of parentheses
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:
Everyone can take part in filling these tables or creating new ones. I insist on one condition though:
Tables may have gaps that would reflect on my current knowledge or cummulative time investment into solving these puzzles. However, I am the only one allowed to create gaps. All others either fill gaps or submit contiguous solutions for a series of numbers. |
- One 4
- 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:
- a - a = 0
- a / a = 1
- (a + a) / a = 2
- (a + a + a) / a = 3
- √a·√a = a
For a digit n there are additional useful identities:
- n / .(n) = 9
- n / .n = 10
- nn / n = 11
(A Java applet that could help experiment with this kind of problems is also avaialble.)
Note: Inder Jeet Taneja has gathered a fantastic collection of various one digit representations of numbers from 1 to 1000.
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