Puzzles
24 December
1,0,2,0,1,1
The sequence of six numbers above has two properties:
- Each number is either 0, 1 or 2.
- Each pair of consecutive numbers adds to (strictly) less than 3.
Today's number is the number of sequences of six numbers with these two properties
23 December
Today's number is the area of the largest area rectangle with perimeter 46 and whose sides are all integer length.
22 December
In base 2, 1/24 is
0.0000101010101010101010101010...
In base 3, 1/24 is
0.0010101010101010101010101010...
In base 4, 1/24 is
0.0022222222222222222222222222...
In base 5, 1/24 is
0.0101010101010101010101010101...
In base 6, 1/24 is
0.013.
Therefore base 6 is the lowest base in which 1/24 has a finite number of digits.
Today's number is the smallest base in which 1/10890 has a finite number of digits.
Note: 1/24 always represents 1 divided by twenty-four (ie the 24 is written in decimal).
21 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 smallest number you can make using the digits in the red boxes.
| + | ÷ | = 2 | |||
| × | + | - | |||
| × | - | = 31 | |||
| + | + | - | |||
| - | × | = 42 | |||
| = 37 | = 13 | = -2 |
20 December
Today's number is the sum of all the numbers less than 40 that are not factors of 40.
19 December
Today's number is the number of 6-dimensional sides on a 8-dimensional hypercube.
18 December
There are 6 terms in the expansion of \((x+y+z)^2\):
$$(x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2xz$$
Today's number is number of terms in the expansion of \((x+y+z)^{16}\).
17 December
For \(x\) and \(y\) between 1 and 9 (including 1 and 9), I write a number at the co-ordinate \((x,y)\): if \(x\lt y\), I write \(x\); if not,
I write \(y\).
Today's number is the sum of the 81 numbers that I have written.