Question

A six digit number (base $10$ ) is squarish if it satisfies the following conditions:

(i) none of its digits is zero;

(ii) it is a perfect square; and

(iii) the first two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.

How many squarish numbers are there?

$0$

$2$

$3$

$8$

$9$

Answer: B

Concepts:digits perfect square systematic listing

Difficulty rating: 2160

Small Hint:

List the two-digit perfect squares having no zero digit

Big Hint:

Use the first pair to restrict the square root to a short interval, then filter by the final pair

Solution:

Each two-digit block must belong to $\{16,25,36,49,64,81\}.$ Restricting the square root by the first block and retaining only squares with an allowed final block leaves the following possible middle blocks:

first block	possible middle blocks
$16$	$32,$ $40,$ $48,$ $56,$ $64,$ $72$
$25$	$40,$ $50,$ $60,$ $70,$ $80,$ $90$
$36$	$48,$ $60,$ $72,$ $84,$ $96$
$49$	$56,$ $70,$ $84,$ $98$
$64$	$64,$ $80,$ $96$
$81$	$72,$ $90$

Only the middle block $64$ is allowed, producing $\begin{aligned} 166464&=408^2,\\ 646416&=804^2. \end{aligned}$ Thus there are $2$ squarish numbers.

Therefore, the correct answer is B.

1980 AMC 12 Problem 30

Answer: B

Difficulty rating: 2160

Small Hint:

Big Hint:

Solution:

Problem 30 in Other Years