356. A simple cryptarithm. Each letter in the following cryptarithm stands for a definite decimal digit:

$$
3(B I D F O R)=4(F O R B I D)
$$

Restore the original record.
356. We can write the following relationship:

$$
3[1000(B I D)+(F O R)]=4[1000(F O R)+(B I D)]
$$

Expanding the square brackets, we get

$$
3000(B I D)+3(F O R)=4000(F O R)+4(B I D)
$$

By combining like terms, we arrive at the relationship $2996(B I D)=$ $=3997(F O R)$. Dividing both sides of this equation by 7, we get the relationship

$$
428(B I D)=571(F O R), \quad \text { or } \quad \frac{428}{571}=\frac{(F O R)}{(B I D)}
$$

Since this fraction has no other form containing three-digit numbers in the numerator and denominator, we find that $B=5$, $I=7, D=1, F=4, O=2, R=8$, and the entire cryptarithm is decoded as follows:

$$
3(571428)=4(428571), \quad \text { or } \quad 1714284=1714284 .
$$

[E. Keogh, M. M., 45, 285 (May 1972).]