Puzzles
22 December
There are 12 ways of placing 2 tokens on a 2×4 grid so that no two tokens are next to each other horizontally, vertically or diagonally:
Today's number is the number of ways of placing 2 tokens on a 2×21 grid so that no two tokens are next to each other horizontally, vertically or diagonally.
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The two tokens must be in two non-adjacent columns. There are ½×21×20 ways of picking two different columns. 20 of these ways will give two adjacent columns,
so there are ½×21×20–20=190 ways to pick the columns.
Once the columns are picked there are four choices for the rows to place the tokens in (up and up, up and down, down and up, down and down). 4×190=760.
21 December
Arrange the digits 1–9 (using each digit exactly once) so that the three digit number in:
the middle row is a prime number;
the bottom row is a square number;
the left column is a cube number;
the middle column is an odd number;
the right column is a multiple of 11.
The 3-digit number in the first row is today's number.
| | | today's number |
| | | prime |
| | | square |
| cube | odd | multiple of 11 |
20 December
What is the area of the largest area triangle that has one side of length 32 and one side of length 19?
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The triangle will be largest when the two sides meet at a right angle. The area of this triangle is ½×32×19=304.
19 December
The equation \(352x^3-528x^2+90=0\) has three distinct real-valued solutions.
Today's number is the number of integers \(a\) such that the equation
\(352x^3-528x^2+a=0\) has three distinct real-valued solutions.
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The function \(f(x)=352x^3-528x^2+a\) with \(a=90\) looks like this (click to enlarge):
Adjusting the value of \(a\) will move the curve up and down. The equation \(f(x)=0\) will have three distinct solutions as long as the local minimum is below the \(x\)-axis
and the local maximum is above the \(x\)-axis.
The minimum and maximum are at \(x=1\) and \(x=0\). The difference between \(f(1)\) and \(f(0)\) is 176. This means that there are 175 possible integer values of \(a\) (as the endpoints are not included).
18 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.
| + | | + | | = 11 |
| + | | × | | × | |
| + | | + | | = 17 |
| × | | - | | + | |
| + | | + | | = 17 |
=
11 | | =
17 | | =
17 | |
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| 6 | + | 3 | + | 2 | = 11 |
| + | | × | | × | |
| 5 | + | 8 | + | 4 | = 17 |
| × | | - | | + | |
| 1 | + | 7 | + | 9 | = 17 |
=
11 | | =
17 | | =
17 | |
The product of the numbers in the red boxes is 432.
17 December
The digital product of a number is computed by multiplying together all of its digits.
For example, the digital product of 6273 is 252.
Today's number is the smallest number whose digital product is 252.
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252 can be written as the product of one-digit numbers in the following ways:
- 2×2×3×3×7
- 2×2×9×7
- 2×3×6×7
- 3×3×4×7
- 6×6×7
- 4×9×7
Therfore, the smallest number whose digital product is 252 is 479.
16 December
Each clue in this crossnumber is formed of two parts connected by a logical connective:
and means that both parts are true;
nand means that at most one part is true;
or means that at least one part is true;
nor means that neither part is true;
xor means that exactly one part is true;
xnor means that either both parts are false or both parts are true.
No number starts with 0.
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1A is a palindrome xnor 1D is a palindrome.
1A is greater than 350 nor 1D is less than 150.
3D is odd nand 4A and 2D are equal.
3D is prime xor 5A is odd.
4A is a cube and 2D is a cube.
The sum of the digits of 3D is 2 or the sum of the digits of 5A is 5.
Today's number is 1D.
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15 December
The odd numbers are written in a pyramid.
| (row 1) | | | 1 | | |
| (row 2) | | 3 | | 5 | |
| (row 3) | 7 | | 9 | | 11 |
| etc. |
What is the mean of the numbers in the 19th row?
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The mean of the numbers in the \(n\)th row is \(n^2\), so the mean of the number in the 19th row is 361.
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