Puzzles
4 December
Today's number is the number of 0s that 611! (611×610×...×2×1) ends in.
3 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 |
| - | | + | | × | |
| + | | - | | = 11 |
| - | | - | | - | |
| + | | + | | = 11 |
=
-11 | | =
11 | | =
11 | |
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| 4 | + | 5 | + | 2 | = 11 |
| - | | + | | × | |
| 8 | + | 9 | - | 6 | = 11 |
| - | | - | | - | |
| 7 | + | 3 | + | 1 | = 11 |
=
-11 | | =
11 | | =
11 | |
The product of the numbers in the red boxes is 108.
2 December
Today's number is the area of the largest dodecagon that it's possible to fit inside a circle with area \(\displaystyle\frac{172\pi}3\).
1 December
There are 5 ways to write 4 as the sum of 1s and 2s:
- 1+1+1+1
- 2+1+1
- 1+2+1
- 1+1+2
- 2+2
Today's number is the number of ways you can write 12 as the sum of 1s and 2s.
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233
Let \(f(n)\) be the number of ways to write \(n\) as the sum of 1s and 2s.
When you write \(n\) as the sum of 1s and 2s, the sum must end in either 1 or 2: if it ends in 1, then there are \(f(n-1)\) ways to write the
rest before this 1 as the sum of 1s and 2s; if it ends in 2, there are \(f(n-2)\) ways to write the rest.
This means that \(f(n)=f(n-1)+f(n-2)\), or in other words the Fibonacci numbers.
Cryptic crossnumber #2
In this puzzle, the clues are written like clues from a cryptic crossword, but the answers are all numbers. You can download a printable pdf
of this puzzle
here.
Across| 1 | Found your far dented horn mixed to make square. | (3) | | 3 | Eno back in Bowie's evening prime. | (2) | | 4 | Prime legs. | (2) | | 5 | Palindrome ends cubone, starts ninetales, inside poison ekans. | (3) |
| | Down| 1 | Odd confused elven elves hounded deerhound antenna. | (5) | | 2 | Prime try of confused Sven with Beckham's second. | (2) | | 4 | Prime even and Ian fed the being. | (2) |
|
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Across
1 Found your far dented horn mixed to make square. (3) — 144 is a square number that is an an anagram of "Found your far dented horn" (one hundred and Forty four).
3 Eno back in Bowie's evening prime. (2) — 17 is a prime that is Eno backwards (onE) followed by a number in Bowie's evening.
4 Prime legs. (2) — 11 is a prime number and called legs in bingo.
5 Palindrome ends cubone, starts ninetales, inside poison ekans. (3) — 191 is a palindrome that is a the end of cubone the start of ninetales and inside poison ekans.
Down
1 Odd confused elven elves hounded deerhound antenna. (5) — 11111 is an odd number that is an anagram of "elven elves hounded deerhound antenna" (eleven thousand one hundred and eleven).
2 Prime try of confused Sven with Beckham's second. (2) — 47 is a prime that is an anagram of "try of" (forty) and Sven with Beckham's second (Seven).
4 Prime even and Ian fed the being. (2) — 19 is a prime number that is the even letters of "and Ian fed the being" (nIneteen).
Cryptic crossnumber #1
In this puzzle, the clues are written like clues from a cryptic crossword, but the answers are all numbers. You can download a printable pdf of this puzzle
here.
Across| 1 | Triangular one then square. | (3) | | 3 | Audible German no between tutus, for one square. | (5) | | 5 | Irreducible ending Morpheus halloumi fix, then Trinity, then mixed up Neo. | (3) |
| | Down| 1 | Inside Fort Worth following unlucky multiple of eleven. | (3) | | 2 | Palindrome two between two clickety-clicks. | (5) | | 4 | Confused Etna honored thundery din became prime. | (3) |
|
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Across
1 Triangular one then square. (3) — 136 is a triangular number that is a one followed by a square number (36).
3 Audible German no between tutus, for one square. (5) — 29241 is a square number that sounds like a German no (nein/nine) between tutus (two two) for one (four one).
5 Irreducible ending Morpheus halloumi fix, then Trinity, then mixed up Neo. (3) — 631 is an irreducible number that is the ending of Morpheus halloumi fix (six), the trinity (three), then mixed up Neo (oNe).
Down
1 Inside Fort Worth following unlucky multiple of eleven. (3) — 132 is a multiple of 11 that is inside Fort Worth (two) following unlucky (13).
2 Palindrome two between two clickety-clicks. (5) — 66266 is a palindrome that is a two between two clickety-clicks (66s).
4 Confused Etna honored thundery din became prime. (3) — 131 is a prime that is an anagram of "Etna honored thundery din" (onE hundred and thirty one).
Breaking Chocolate
You are given a bar of chocolate made up of 15 small blocks arranged in a 3×5 grid.
You want to snap the chocolate bar into 15 individual pieces. What is the fewest number of snaps that you need to break the bar? (One snap consists of picking up one piece of chocolate and snapping it into two pieces.)
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Each snap increases the total number of pieces by one. So in order to make 15 pieces, you will need to perform 14 snaps.
Square and cube endings
Source: UKMT 2011 Senior Kangaroo
How many positive two-digit numbers are there whose square and cube both end in the same digit?
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Only the units digit of the number will affect the last digit of the square and cube. This table shows how the last digits of the square and cube depend on the last digit of the number:
| Last digit of... |
| number | square | cube |
| 0 | 0 | 0 |
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 7 |
| 4 | 6 | 4 |
| 5 | 5 | 5 |
| 6 | 6 | 6 |
| 7 | 9 | 3 |
| 8 | 4 | 2 |
| 9 | 1 | 9 |
So numbers ending in 0, 1, 5 and 6 will have squares and cubes that end in the same digit. There are 4×9=36 two-digit numbers then end in one of these digits.
Extension
How many two-digit numbers are there in binary whose square and cube end in the same digit?
How many two-digit numbers are there in ternary whose square and cube end in the same digit?
How many two-digit numbers are there in base \(n\) whose square and cube end in the same digit?
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