Coding/decoding

[Kieran]

A code contains 7 digits. Each number in the code is greater than the previous number. The sum of the digits is a perfect square. This is the largest 7 digit number where both constraints are obeyed. What is the number ?
There is no arithmetic or geometric relation between the numbers, this makes it more difficult. The first 3 numbers could be a, a+p, a + p +q. A lot of unknowns. Thanks.

 

[Deleted member 4993]

A code contains 7 digits. Each number in the code is greater than the previous number. The sum of the digits is a perfect square. This is the largest 7 digit number where both constraints are obeyed. What is the number ?
There is no arithmetic or geometric relation between the numbers, this makes it more difficult. The first 3 numbers could be a, a+p, a + p +q. A lot of unknowns. Thanks.

Without the second constraint - the largest number the code can contain = 3456789

sum of the digits = 42

Without the second constraint - the smallest number the code can contain = 0123456

sum of the digits = 21

How many perfect squares are there that are smaller than 42 but larger than 21?

 

[soroban]

Hello, Kieran!

Subhotosh was that close to the answer . . .

A code contains 7 digits.
Each digit in the code is greater than the previous digit.
The sum of the digits is a perfect square.
This is the largest 7-digit number where both constraints are obeyed.
What is the number?

The largest 7-digit number is \(\displaystyle 3456789.\)
But the digit sum is \(\displaystyle 42.\)
We must reduce the digit sum by 6.

The new number is: \(\displaystyle 2345679.\)