Puzzles

The product of the red digits is 192.

Carol will get 270 whatever she does.

The only way to make 208 by adding consecutive numbers is 10+11+12+13+14+15+16+17+18+19+20+21+22. Therefore next year Noel will give his grandchildren a total of £221.

Let \(a_n\) be the number of ways to tile an \(n\times2\) rectangle.

There is one way to tile a 1×2 rectangle; and there are three ways to tile a 2×2 rectangle. Therefore \(a_1=1\) and \(a_2=3\).

In a \(n\times2\) rectangle, the rightmost column will either contain a vertical 2×1 domino, two horizontal 2×1 dominoes, or a 2×2 square. Therefore \(a_n=a_{n-1}+2a_{n-2}\).

Using this, we find that there are 341 ways to tile a 9×2 rectangle.

The largest number you can make with the digits in the red boxes is 432.

In order to make a valid triangle, the sum of the two shorter sides must be larger than the longest side. There is 1 triangle with longest side 4cm (2,3,4); 2 with longest side 5cm (2,4,5 and 3,4,5); 4 with longest side 6cm (2,5,6; 3,5,6; 4,5,6 and 3,4,6); 6 with longest side 7cm (2,6,7; 3,6,7; 4,6,7; 5,6,7; 3,5,7 and 4,5,7); 9 with longest side 8cm; 12 with longest side 9cm; 16 with longest side 10cm; 20 with longest side 11cm; 25 with longest side 12cm; 30 with longest side 13cm; 36 with longest side 14cm; and 42 with longest side 15cm.

In total this makes 203 triangles.

100 numbers will have 1 in the units column. Another 100 with have a 1 in the tens column; and other 100 will have a 1 in the hundreds column. Finally, 1000 has a 1 in the thousands column.

In total, this makes 301 copies of the digit 1.