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# Puzzles

## 24 December

There are six ways to put two tokens in a 3 by 3 grid so that the diagonal from the top left to the bottom right is a line of symmetry:
Today's number is the number of ways of placing two tokens in a 29 by 29 grid so that the diagonal from the top left to the bottom right is a line of symmetry.

## 23 December

198 is the smallest number that is equal to 11 times the sum of its digits.
Today's number is the smallest number that is equal to 48 times the sum of its digits.

## 22 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 largest number you can make with the digits in the red boxes.
 + + = 18 + + + ÷ - = 1/2 + + + + ÷ = 3/2 =24 =8 =13
Tags: numbers, grids

## 21 December

There are 3 ways to order the numbers 1 to 3 so that no number immediately follows the number one less that itself:
• 3, 2, 1
• 1, 3, 2
• 2, 1, 3
Today's number is the number of ways to order the numbers 1 to 6 so that no number immediately follows the number one less that itself.

## 20 December

18 can be written as the sum of 3 consecutive (strictly) positive integers: 5 + 6 + 7.
18 can also be written as the sum of 4 consecutive (strictly) positive integers: 3 + 4 + 5 + 6.
18 is in fact the smallest number that can be written as the sum of both 3 and 4 consecutive (strictly) positive integers.
Today's number is the smallest number that can be written as the sum of both 12 and 13 consecutive (strictly) positive integers.
Tags: numbers, sums

## 19 December

The diagram to the right shows a triangle. Two of the sides of the triangle have been split into three pieces, with lines drawn from the opposite vertex. In total, the diagram now contains 27 triangles of any size.
Another triangle has two of its sides split into eight pieces, with lines drawn from the opposite vertex. How many triangles (of any size) would this create?

## 18 December

The expansion of $$(x+y+z)^3$$ is
$$x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3xy^2 + 3y^2z + 3xz^2 + 3yz^2 + 6xyz.$$
This has 10 terms.
Today's number is the number of terms in the expansion of $$(x+y+z)^{26}$$.
Tags: algebra

## 17 December

Put the digits 1 to 9 (using each digit exactly once) in the boxes so that the sums and product are correct. Today's number is the product of the numbers in the red boxes.
 + + = 16 + + + + + = 8 + + + × × = 288 =11 =14 =20
Tags: numbers, grids

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