000RM.dvi

13.5 The problem of 4n’s 411

10.Here are seven consecutive squares for each of which its decimal

digits sum to a square:

81 , 100 , 121 , 144 , 169 , 196 , 225.

Find another set of seven consecutive squares with the same prop-

erty.^1

11.Find a perfect square of 12 digits formed from the juxtaposition of

two squares, one having 4 digits and the other 8 digits.

12.A pandigital number is one whose decimal representation contains

all digits 0, 1,..., 9. There are three pandigital perfect squares

whose square roots are palindromes. Find them.

13.Find the smallest 3-digit numberN such that the three numbers

obtained by cyclic permutations of its digits are in arithmetic pro-

gression.

14.Form a square of 8 digits which is transformed into a second square

when the second digit from the left is increased by 1.

15.The number(abbbb)^2 − 1 has 10 digits, all different. Find the num-

ber.^2

1 The seven numbers beginning with 9999.
2. 1 = 9560341728−^297777 ; 1 = 7319658024−^285555 These are the only possibilities even if we
consider more generally numbers consisting of two consecutive blocks of repeating digits, whose squares, to
within± 1 , contain all ten digits without repetition.