This solution comes from Andrei from School No. 205, Bucharest,
Romania.
To solve this problem I followed the following steps:
- associating to the letters of the alphabet numbers between 0
and 25, I transformed the coded message into a set of pairs of
numbers $(\alpha',\beta')$
- I solved the system of equations for $(\alpha,\beta)$ in
terms of $(\alpha',\beta')$.
- I used the same association as in the first step, and I
transformed the set of numbers $(\alpha, \beta)$ into letters,
and I found the message.
| a |
b |
c |
d |
e |
f |
g |
h |
i |
j |
k |
l |
m |
n |
o |
p |
q |
r |
s |
t |
u |
v |
w |
x |
y |
z |
| 0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
21 |
22 |
23 |
24 |
25 |
For each pair of numbers $(\alpha', \beta')$ I have to solve
the system to determine $(\alpha, \beta)$ $$ \alpha'= \alpha +
3\beta\pmod {26}\quad (1) $$ $$\beta' = 5\beta \quad
\pmod{26}\quad (2)$$
I start from the last equation: $$\beta = {1\over 5}\beta'
\pmod{26}$$ To determine 1/5 (mod 26), I first constructed the
table of multiplication for 5 (mod 26) to see where I obtain 1.
As 21 multiplied by 5 gives 1 (mod 26) it follows that 1/5 (mod
26) is 21. This means: $$\beta = 21\beta' \pmod{26}\quad (3)$$
and for $\alpha$ I obtained successively $$\alpha =
\alpha'-3\beta = \alpha'-3\times 21\beta' = \alpha' - 11\beta'
\pmod{26} $$ that is $$\alpha = \alpha' +15\beta' \pmod{26}
\quad (4).$$ Now, the sequence of numbers $(\alpha', \beta')$
is transformed by equations (3) and (4) into the sequence
$(\alpha,\beta)$
| $C$ |
$\alpha'$ |
$\beta'$ |
$\alpha$ |
$\beta$ |
$P$ |
| dj |
3 |
9 |
8 |
7 |
ih |
| lb |
11 |
1 |
0 |
21 |
av |
| rn |
17 |
13 |
4 |
13 |
en |
| qm |
16 |
12 |
14 |
18 |
os |
| bu |
1 |
20 |
15 |
4 |
pe |
| ao |
0 |
14 |
2 |
8 |
gi |
| hd |
7 |
3 |
0 |
11 |
al |
| eo |
4 |
14 |
6 |
8 |
gi |
| kr |
10 |
17 |
5 |
19 |
ft |
| ia |
8 |
0 |
8 |
0 |
ia |
| cs |
2 |
18 |
12 |
14 |
mo |
| ud |
20 |
3 |
13 |
11 |
nl |
| rx |
17 |
23 |
24 |
15 |
yp |
| cm |
2 |
12 |
0 |
18 |
as |
| qo |
16 |
14 |
18 |
8 |
si |
| bn |
1 |
13 |
14 |
13 |
on |
| fr |
5 |
17 |
0 |
19 |
at |
| ld |
11 |
3 |
4 |
11 |
el |
| ek |
4 |
10 |
24 |
2 |
yc |
| th |
19 |
7 |
20 |
17 |
ur |
| ys |
24 |
18 |
8 |
14 |
io |
| wm |
22 |
12 |
20 |
18 |
us |
The message could be read as the quotation from Einstein
talking about himself as a mathematician:'I have no special
gift. I am only passionately curious'.