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Puzzles

25 December

It's nearly Christmas and something terrible has happened: there's been a major malfunction in multiple machines in Santa's toy factory, and not enough presents have been made. Santa has a backup warehouse full of wrapped presents that can be used in the case of severe emergency, but the warehouse is locked. You need to help Santa work out the code to unlock the warehouse so that he can deliver the presents before Christmas is ruined for everyone.
The information needed to work out the code to the warehouse is known by Santa and his three most trusted elves: Santa is remembering a three-digit number, and each elf is remembering a one-digit and a three-digit number. If Santa and the elves all agree that the emergency warehouse should be opened, they can work out the code for the door as follows:
But this year, there is a complication: the three elves are on a diplomatic mission to Mars to visit Martian Santa and cannot be contacted, so you need to piece together their numbers from the clues they have left behind:
You can to open the door here.

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24 December

There are 343 three-digit numbers whose digits are all 1, 2, 3, 4, 5, 6, or 7. What is the mean of all these numbers?

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23 December

In a grid of squares, each square is friendly with itself and friendly with every square that is horizontally, vertically, or diagonally adjacent to it (and is not friendly with any other squares). In a 5×5 grid, it is possible to colour 8 squares so that every square is friendly with at least two coloured squares:
It it not possible to do this by colouring fewer than 8 squares.
What is the fewest number of squares that need to be coloured in a 23×23 grid so that every square is friendly with at least two coloured squares?

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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 that can be formed using the three digits in the red boxes.
+÷= 3
× ×
+= 6
××= 16
=
1
=
13
=
16

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Tags: numbers, grids

21 December

Noel wants to write a different non-zero digit in each of the five boxes below so that the products of the digits of the three-digit numbers reading across and down are the same.
What is the smallest three-digit number that Noel could write in the boxes going across?

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20 December

p(x) is a polynomial with integer coefficients such that:
What is p(23)?

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19 December

There are 9 integers below 100 whose digits are all non-zero and add up to 9: 9, 18, 27, 36, 45, 54, 63, 72, and 81.
How many positive integers are there whose digits are all non-zero and add up to 9?

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18 December

If k = 21, then 28k ÷ (28 + k) is an integer.
What is the largest integer k such that 28k ÷ (28 + k) is an integer?

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