CF 1521A - Nastia and Nearly Good Numbers

Rating: 1000
Tags: constructive algorithms, math, number theory
Solve time: 2m 11s
Verified: no

Solution

Setup

A radix conversion routine transforms a nonnegative integer written in one base into its representation in another base. The program under discussion is Method 1a from Section 4.4, which produces the digits of an integer $u$ in radix $B$ by repeated division by $B$ in the original radix $b$. Concretely, it generates digits $U_0, U_1, U_2, \ldots$ such that

$$u = (\ldots U_2 U_1 U_0)_B,$$

where each digit is obtained by successive quotient and remainder operations

$$U_0 = u \bmod B,\quad U_1 = \lfloor u/B \rfloor \bmod B,\quad \ldots$$

until the quotient becomes zero.

The exercise asks to justify that a MIX implementation of this method remains valid when the constant $10$ in the division instruction is replaced by a more general constant $c$, provided the arithmetic interpretation of the program is preserved.

The essential claim is that the algorithm does not depend on the specific decimal base, but only on the correctness of the quotient-remainder decomposition

$$u = qB + r,\quad 0 \le r < B.$$

Solution

The program in equation (1) performs repeated division by $B = 10$, extracting the remainder each time as the next radix-$10$ digit. The key invariant maintained across iterations is that at the start of each loop, the register pair $(rA, rX)$ stores the current value $u^{(j)}$, and after executing the division step, this value is decomposed as

$$u^{(j)} = 10 q^{(j)} + r^{(j)}, \quad 0 \le r^{(j)} < 10,$$

so that $r^{(j)}$ is stored as the next output digit and $q^{(j)}$ becomes the next value to process.

The correctness argument uses only the property that division in MIX produces quotient and remainder satisfying

$$rAX = Bq + r,\quad 0 \le r < B.$$

If the instruction DIV =10= is replaced by DIV =c=, the machine computes instead

$$rAX = cq + r,\quad 0 \le r < c,$$

so the loop now extracts digits in base $c$ rather than base $10$. Every step of the algorithm continues to maintain the same structural invariant: after $j$ iterations, the value stored in the working registers equals

$$\left\lfloor \frac{u}{c^j} \right\rfloor,$$

and the emitted remainders are exactly

$$U_j = \left\lfloor \frac{u}{c^j} \right\rfloor \bmod c.$$

Since radix representation is defined uniquely by repeated extraction of remainders in this form, the program computes the base-$c$ expansion of $u$.

No other part of the program depends on the constant $10$ except as the divisor used in this quotient-remainder decomposition. Therefore substituting any fixed integer $c \ge 2$ preserves correctness, and only changes the radix of the output representation.

This completes the proof. ∎

Verification

The correctness depends on two independent conditions that must both hold in MIX arithmetic. First, the DIV instruction must implement exact Euclidean division, producing quotient and remainder satisfying $0 \le r < c$. This ensures that each extracted digit lies in the correct digit range for base $c$.

Second, the recurrence relation induced by repeated division must reduce the magnitude of the working value at each step. Since each iteration replaces $u$ by $\lfloor u/c \rfloor$, we have

$$\left\lfloor \frac{u}{c^{j+1}} \right\rfloor < \left\lfloor \frac{u}{c^j} \right\rfloor$$

whenever the value is nonzero, guaranteeing termination after finitely many steps.

The digit reconstruction identity

$$u = \sum_{j \ge 0} U_j c^j$$

follows directly by unwinding the recurrence for repeated quotient extraction, identical to the standard base-$10$ proof with the constant replaced everywhere by $c$.

Notes

The constant $10$ in Method 1a plays no structural role beyond defining the radix. The same program template therefore implements conversion to any fixed radix $c$ available as a machine constant in the division instruction. The only restriction is that MIX division must behave according to Euclidean division semantics, ensuring $0 \le r < c$, which guarantees valid digit extraction.