Re: Enigma 1504 - All ten digits

On 10 Sep, 09:04, Chappy <petergregorychap...@xxxxxxxxxxx> wrote:
Enigma 1504 - All ten digits
New Scientist magazine, 26 July 2008.
By Richard England.

Harry has chosen some 5-digit perfect
squares (all different) which use only the
digits 0 to 4, none of them starting or
finishing with the digit 0.  Each of the
digits 0 to 4 is used a different number of
times in his squares, those numbers of times
being 5 to 9.

(1) Which eligible square or squares has
    Harry NOT chosen?

Tom has chosen some 5-digit perfect squares
which only use the digits 5 to 9.  Each of
the digits 5 to 9 is used a different number
of times in his squares, those numbers of
times being 0 to 4.

(2) Which squares has Tom chosen?

Ciao,
Chappy.

A.

Start by generating all the valid 5 digit squares that do not start or
end with 0 and contain only the digits 0, 1, 2, 3 or 4....

10201
10404
12321
22201
23104
32041
33124
40401

3 is only present 5 times, so all numbers with 3 in them must be
present (12321,23104,32041,33124).
This gives us three 4s - since the only other numbers have two 4s in
them, and we already have five 3s, both 40401 and 10404 must also be
included.
Add 10201 to the list gives five 3s, six 2s, seven 4s, eight 0s and
nine 1s, so the only digit not used is 22201.

B.

Again, generate valid 5 digit squares that only contain digits 5, 6,
7, 8 or 9....

55696
69696
97969
98596
99856

We are only going to be able to use 2 numbers (1 + 2 + 3 + 4 = 10
digits = 2 five digit numbers). It is relatively trivial to find the
two numbers used are 55696 and 97969
.

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