CF 1514A - Perfectly Imperfect Array

Rating: 800
Tags: math, number theory
Solve time: 2m 51s
Verified: no

Solution

Method 1b already provides the key idea. If

$(u_m\ldots u_1u_0)_3$

is a ternary integer, its decimal value is obtained from the nested form

$$(\cdots((u_m\cdot3+u_{m-1})\cdot3+u_{m-2})\cdots)\cdot3+u_0.$$

For pencil-and-paper work this means:

Start with the leading digit. Repeatedly multiply the current result by $3$ and add the next ternary digit.

Only multiplication by $3$ and addition of $0$, $1$, or $2$ are required.

For

$$(12120112120210)_3,$$

the computation proceeds as follows:

$$\begin{array}{rcl} 1 &\to& 1\cdot3+2=5,\ 5 &\to& 5\cdot3+1=16,\ 16 &\to& 16\cdot3+2=50,\ 50 &\to& 50\cdot3+0=150,\ 150 &\to& 150\cdot3+1=451,\ 451 &\to& 451\cdot3+1=1354,\ 1354 &\to& 1354\cdot3+2=4064,\ 4064 &\to& 4064\cdot3+1=12193,\ 12193 &\to& 12193\cdot3+2=36581,\ 36581 &\to& 36581\cdot3+0=109743,\ 109743 &\to& 109743\cdot3+2=329231,\ 329231 &\to& 329231\cdot3+1=987694,\ 987694 &\to& 987694\cdot3+0=2963082. \end{array}$$

Hence

$$(12120112120210)3=(2963082){10}.$$

Therefore the decimal value is

$$\boxed{2963082}.$$

To convert in the opposite direction, from decimal to ternary, use Method 1a with $B=3$. Repeatedly divide by $3$ and record the remainders.

For an integer $u$,

$$u=3q_0+r_0,\qquad 0\le r_0<3,$$

then

$$q_0=3q_1+r_1,$$

and so on until the quotient becomes $0$. The ternary digits are the remainders read in reverse order:

$$u=(\ldots r_2r_1r_0)_3.$$

Thus a rapid ternary-to-decimal method is repeated multiplication by $3$ and addition of the next digit; a rapid decimal-to-ternary method is repeated division by $3$ with remainders.

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