Puzzles
21 December
There are 6 two-digit numbers whose digits are all 1, 2, or 3 and whose second digit onwards
are all less than or equal to the previous digit:
- 33
- 22
- 31
- 21
- 11
- 32
How many 20-digit numbers are there whose digits are all 1, 2, or 3 and whose second digit onwards
are all less than or equal to the previous digit?
19 December
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 numbers in the red boxes.
| + | – | = 7 | |||
| × | × | × | |||
| + | – | = 0 | |||
| ÷ | ÷ | ÷ | |||
| + | – | = 2 | |||
| = 4 | = 35 | = 18 |
18 December
Some numbers can be written as the product of two or more consecutive integers, for example:
$$6=2\times3$$
$$840=4\times5\times6\times7$$
What is the smallest three-digit number that can be written as the product of two or more consecutive integers?
15 December
The arithmetic mean of a set of \(n\) numbers is computed by adding up all the numbers, then
dividing the result by \(n\).
The geometric mean of a set of \(n\) numbers is computed by multiplying all the numbers together, then
taking the \(n\)th root of the result.
The arithmetic mean of the digits of the number 132 is \(\tfrac13(1+3+2)=2\).
The geometric mean of the digits of the number 139 is \(\sqrt[3]{1\times3\times9}\)=3.
What is the smallest three-digit number whose first digit is 4 and for which the arithmetic and geometric means of its digits are both non-zero integers?
12 December
What is the smallest value of \(n\) such that
$$\frac{500!\times499!\times498!\times\dots\times1!}{n!}$$
is a square number?
11 December
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 numbers in the red boxes.
| + | + | = 15 | |||
| + | + | ÷ | |||
| + | – | = 10 | |||
| + | – | × | |||
| ÷ | × | = 3 | |||
| = 16 | = 1 | = 30 |
10 December
How many integers are there between 100 and 1000 whose digits add up to an even number?
8 December
Noel writes the numbers 1 to 17 in a row. Underneath, he writes the same list without the first and last numbers, then continues this until he writes a row containing just one number:
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | ||
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||||
| etc. | ||||||||||||||||
What is the sum of all the numbers that Noel has written?