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What is the smallest perfect square that ends with the four digits 9009?

Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Mod 7

Find the remainder when 3^{2001} is divided by 7.

Double Time

Age 16 to 18
Challenge Level
To encode the following message the text was first written as pairs of letters ignoring the spaces between the words. For example John Brown would be written jo hn br ow nz where the z is added because there are an odd number of letters.

The letters of the alphabet are numbered $a=0,\ b=1,\ {\rm to}\ z=25$ and each pair of letters is replaced by a pair of numbers. There are $26\times 26=676$ pairs of numbers. It is possible to store the codes for all 676 pairs and look them up as needed but much more efficient to use a decyphering formula.

Each pair of numbers $(\alpha, \beta)$ is encoded as another pair of numbers $(\alpha',\beta')$ where $$\alpha' = \alpha + 3\beta \pmod {26}$$ $$\beta' = 5\beta \ \ \pmod{26}$$ Find $\alpha$ and $\beta$ in terms of $\alpha'$ and $\beta'$ and hence decode the following quotation which is a remark made by Einstein:

dj lb rn qm bu ao hd eo kr ia cs ud rx cm qo bn fr ld ek th ys wm