Puzzles
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?
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?
Show answer & extension
Hide answer & extension
All the integers whose digits are all non-zero and add up to 9 can be made by taking a list of 9 ones, and between each pair deciding whether to move on a digit or not. For example,
1111|111|11 (where | means move on) would represent the number 432; and 1|1|1111111 would represent the number 117.
There are 8 gaps between 9s, so there are 28 = 256 numbers that can be made.
Extension
Why are there not 512 positive integers whose digits are all non-zero and add up to 10?
15 December
The number 2268 is equal to the product of a square number (whose last digit is not 0) and the same square number with its digits reversed:
36×63.
What is the smallest three-digit number that is equal to the product of a square number (whose last digit is not 0) and the same square number with its digits reversed?
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Hide answer
The only three-digit number that is equal to the product of a square number and the same square number with its digits reversed is 16×61 = 976.
14 December
153 is 3375. The last 3 digits of 153 are 375.
What are the last 3 digits of 151234567890?
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Hide answer
We can look at the first few powers and look for a pattern:
| n | 15n | Last 3 digits of 15n |
| 1 | 15 | 15 |
| 2 | 225 | 225 |
| 3 | 3375 | 375 |
| 4 | 50625 | 625 |
| 5 | 759375 | 375 |
| 6 | 11390625 | 625 |
| 7 | 170859375 | 375 |
| 8 | 2562890625 | 625 |
| 9 | 38443359375 | 375 |
If n is even (and > 2), the last three digits of 15n are 625.
12 December
Holly picks a three-digit number. She then makes a two-digit number by removing one of the digits.
The sum of her two numbers is 309. What was Holly's original three-digit number?
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Holly's sum is odd, so she must have removed the units digit and so her calculation was:
$$
\begin{array}{cccc}
a&b&c\\
&a&b&+\\
\hline
3&0&9
\end{array}
$$
\(a\) and \(b\) cannot both be 0, so the must sum in the tens column must've caused a carry into the hundreds column. This means that \(a\) must be 2, and the calculation is:
$$
\begin{array}{cccc}
2&b&c\\
&2&b&+\\
\hline
3&0&9
\end{array}
$$
Two single digits cannot add to 19, so there can't be a carry from the units column into the tens column. This means that \(b\) is 8:
$$
\begin{array}{cccc}
2&8&c\\
&2&8&+\\
\hline
3&0&9
\end{array}
$$
We can see now that \(c\) was 1, so Holly's three digt number was 281.
6 December
The number n has 55 digits. All of its digits are 9.
What is the sum of the digits of n3?
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Hide answer
We can look for a pattern as we increate the number of 9s that make up n:
| n | n3 |
| 9 | 729 |
| 99 | 970299 |
| 999 | 997002999 |
| 999 | 999700029999 |
| 9999 | 999970000299999 |
If n has k digits, then n3 is
k-1 9s,
followed by a 7,
followed by k-1 0s,
followed by a 2,
followed by k 9s,
The sum of all these digits will by 18k.
Hence, the answer is 18×55 = 990.
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:
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?
Show answer & extension
Hide answer & extension
We can look at how many \(n\)-digit number there are for small values of \(n\) and look for a pattern:
- 1-digit numbers: there are 3.
- 2-digit numbers: there are 6.
- 3-digit numbers: there are 10.
- 4-digit numbers: there are 15.
These are the triangle numbers, and there are 231 20-digit numbers.
Extension
Why is the pattern the triangle numbers?
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?
Show answer & extension
Hide answer & extension
The only three digit number whose first digits is 4 and for which the arithmetic and geometric means of its digits are both non-zero integers is 444.
Extension
How many three digit numbers are there for which the arithmetic and geometric means of its digits are both non-zero integers?
How many four digit numbers are there for which the arithmetic and geometric means of its digits are both non-zero integers?
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