Sunday Afternoon Maths LXV
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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