Puzzles

In July, I posted the Combining Multiples puzzle.

Today's number is the largest number that cannot be written in the form \(27a+17b\), where \(a\) and \(b\) are positive integers (or 0).

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the smaller number in a red box to the power of the larger number in a red box.

Here is a list of facts about today's number:

The smallest number that is equal to the sum of its digits multiplied by ten more than the sum of its digits.

Today's number is the second smallest number that can be written as a×b×c×d×e×f×g×h×i, where a,b,...,i are all integers greater than 1.

Put the digits 1 to 9 (using each digit once) in the boxes so that the three digit numbers formed (reading left to right and top to bottom) have the desired properties written by their rows and columns.

When you add up the digits of a number, the result is called the digital sum.

How many different digital sums do the numbers from 1 to 10⁹¹ have?*

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums are correct. The sums should be read left to right and top to bottom ignoring the usual order of operations. For example, 4+3×2 is 14, not 10. Today's number is the product of the digits in the red boxes.