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'Double Time' printed from https://nrich.maths.org/
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'.