Puzzles

29 is odd, and the difference between consecutive square numbers is always odd so it makes sense to check if the relevant two consecutive squares have three digits: these squares are 196 and 225.

The square numbers must be 1, 9, 25, 36, and 784.

The sets of numbers between 1 and \(n\) are either also valid sets with numbers between 1 and \(n-1\); or they can be made by taking a valid set of numbers between 1 and \(n-2\), adding 1 to each number and appending an \(n\) to the set. Therefore the number of sets is the sum of the previous two terms (ie it's the Fibonacci sequence).

Using this, we can work out that the number of sets of numbers between 1 and 14 is 377.

The final digit is 1

Once the leftmost digit of a row and the row below it are decided, there is only one way to fill the rest of the row. Hence, there are two options for the next row (0 0 or 1 1), and four options for the next row (the rows starting 0 and 1 for both 0 0 and 1 1), and in general there are double the number of options for each row as we go upwards.

Therefore for the row with 10 digits there are 2⁹ = 512 options.

The product is 120.

In order for the number of odd factors of the number to be odd, it must be an odd square number or a power of two times an odd square number. 128 is 1 (an odd square) times 128 (a power of two) and no smaller three digit number is of this form.

A valid sequence of length \(n\) can be made by either adding a single B to the right of a valid sequence of length \(n-1\) or two As to a valid sequence of length \(n-2\). Therefore, the number of sequences of length \(n\) can be calculated by adding the previous two terms (ie it's the Fibonacci sequence).

Using this, we can work out that the number of sequence of length eleven is 144.

3 down is a multiple of 101, so its middle digit must be 0:

The clue 4 across now tells us that the units digit of today's number (ie 1 across) is 5 or 0. As 3 down cannot begin with 0, it must be 5.

3 down is a multiple of 101:

The sum of the digits of today's number is at most 9+9+5 = 23, so 1 down starts with a 1 or 2. This means that the sum of the digits of today's number is at most 2 + 9 + 5 = 16, so 1 down must start with a 1:

Two times today's number (ie 4 across) must start with a 2 or 3. If it started with 2, then the sum of the digits of today's number (1 down) would be 12, so today's number would be 165. The product of the digits of today's number (5 across) would be 30. This is not possible as 5 across ends in 5. The first digit of 4 across must therefore be a 3:

The sum of the digits of today's number is 13, and we can fill in the rest using the clues for 4 and 5 across:

Therefore, today's number is 175.