how can we make numbers from 1 to 15 with three 2? (you can make this numbers only with+,-,*,/,√,⌊…⌋,⌈…⌉,...^...) e. g. :6=2+2+2 3=(2+2)-⌊2√⌋
This classic style problem is typically stated in a form similar to the following:
"Represent all integers using four 4's and a finite number of mathematical symbols and/or operators in common use."
The groundrules and assumptions usually posed:
* Cannot use representations for other numbers, eg: use of [[pi]] in sin([[pi]]/4) = 1
* Cannot use ..... to represent an infinite number of operations eg: sqrt(sqrt(sqrt(...4))) = 1
* Obviously negative integers can be represented if a positive number can.
* Acceptable symbols and operators
.........+ for plus, - for minus
.........* for multiplication, / for division
.........sqrt for square root, ^ for raised to the power of.
.........! for factorial, eg: n! = n * (n-1) * (n-2) * ... * 2 * 1
........- . for decimal point, eg: .4
........` for repeating decimal, eg: .4` = .444444444.....= 4/9
In situations where more than one solution can be found, the challenge is to find the simplest, using the least number of symbols and operators, as well as the simplest operators. Operator simplicity in increasing order is +, - , *, /, sqrt,^, !, decimal point, and finally the repeating decimal symbol `.
Subfactorial
Though not too well known, or used for that matter, the subfactorial is a candidate for use in the solutions. The subfactorial is defined as follows: !n = n! (1 - 1/1! + 1/2! - 1/3! + 1/4! - ... +/-1/n!) = n!E where E = the first (n+1) terms of {e^x} for x = -1.
As examples: !1=0; !2=1; !3=2; !4=9; !5=44; !6=265.
While subfactorials are not used that often, it is a valid mathematical term but not always accepted in solutions to the four 4's problem.
Gamma function
An aeronautical engineer from Culver City, California, suggested the use of the Gamma, or generalized factorial function, expressed by Gamma(n) = (n - 1)!. It can be used to get a single four expression of 6, i.e., G(4) = (4 - 1)! = 3! = 6. The Gamma function, while valid, is also not always accepted in solutions.
Some typical solutions for 1 - 10 are given here.
0---4/4 - 4/4
1---4x4/4x4
2---4/4 + 4/4
3---4 - srqt4 + 4/4
4---sqrt(4/.4`) + 4/4
5---sqrt(4/.4`) + 4/sqrt4
6---4!x4/4x4
7---4 + 4 - 4/4
8---4 + 4 + 4 - 4
9----4 + 4 + 4/4
10---4 + 4 + 4/sqrt4
Some excellent additional information on this interesting topic may be found in the following:
1--Mathematical Recreations and Essays by W.W. Rouse Ball & H.S.M. Coxeter, (Dover Publications, Inc., 1987).
2--Mathematical Bafflers by Angela Dunn, Dover Publications, Inc., 1980, pp. 3-8, provides another table of 1 - 100 made from four 4's.
3--The Surprise Attack in Mathematical Problems by L.A. Graham, Dover Publications, Inc., 1968, pp. 27-28.
4-- Fun With Figures by L. Harwood Clarke (William Heinemann Ltd., 1954), pp.51-53, presents a table of solutions for 1 - 100.
5-- 536 Curious Problems & Puzzles by Henry E. Dudeney, Edited by Martin Gardner, (Barnes & Noble Books, 1995)
Using three 2's:
1 - (2/2)^2
2 - 2(2/2)
3 - 2 + (2/2)
4 - (2x2)(!2)
5 - (2+2)(!2)
6 - 2+2+2
7 -
8 - 2x2x2
9 -
10 -
11 - 22/2
12 - (2^2)!/2
13 -
14 -
15 -
16 - (2x2)^2
You probably have these easier ones.