- Write out the multiplication table a×b for a and b from 1 to 2.
What is the sum of all the products in the table?
What is the sum if we have rows for a and columns for b from 1 to 3? ANd what about a 4-by-4 multiplcation table?
Guess a formula for the product's total for n rows and n columns.
What is the sum for n=9?
Answer
1+2+2+4=91 2 2 4
=32
1+2+3 + 2+4+6 + 3+6+9 = 361 2 3 2 4 6 3 6 9
= 62n rows by n columns
The first row is 1+2+...+n = Tri(n) = n(n+1)/2
The second row is 2+4+...+2n = 2 Tri(n) = m(n+1)
...
The last row is n+2n+...+n×n = n Tri(n) = n n(n+1)/2
the sum of all rows is (1+2+...+n)Tri(n) = Tri(n)2The sum of the products in a 9x9 table is Tri(9)2 = 452 = 2025
- How many rectangles of any shape (including squares) can you draw using the lines of a grid which is n squares by n squares? Try it for 1 square, a grid of 2x2, of 3x3.
How many possible rectangles are there in a 9x9 grid of squares?
Answer
A 1x1 grid has just 1 square: total 1A 2x2 grid of squares has 4@1x1, 2@1x2, 2@2x1, 1@2x2: total 9=32A 3x3 grid of squares has 9@1x1, 6@1x2, 6@2x1, 4@2x2, 3@1x3, 3@3x1,
2@2x3, 2@3x2, 1@4x4:total 36=62In general there are Tri(n)2 possible rectangles. See A000537
In a 9x9 grid of squares there are Tri(9)2 = 452 = 2025
Binary:
- 2025 is the square of 45 and 45 is 1011012 in binary and a palindrome.
Find the other square numbers n2 smaller than 2025 where n is also a binary palindrome.
Runs
For more on Runs of natural numbers and their sums (runsums), see Introducing Runsums.
For sums of odd numbers see Runs of Odd Numbers- What are the starting and ending numbers of the run of n natural numbers centered on n when n is odd?
For example: 3+4+5+6+7 is a run of length 5 centred on the number 5, starting at 3 and ending with 7.
What is special about their sums?
Answer
For a run of 2k+1 natural numbers centered on 2k+1, the first is k+1 and the last is 3k+1 with a sum of (2k+1)²
The sums are the square of (the odd number) n - Find a formula for the sum of n numbers beginning at n. The list of these sums starts:
1, 5, 9, 18, 25, 39, A110349.
Answer
if the starting number is odd: 2n-1 + 2n + ... + 2n+(2n-2) = (2n-1)²
If the starting number is even: 2n + 2n+1 + ... 2n+(2n-1) = 4n² + n - Find all the runs of odd numbers with a sum of 2025.
(Hint: See the sums of odd numbers link above for an easy method to do this.)
Answer
1: 2025 =1+3+...+87+89 ≡ 1⊕89 ≡ 45⊗45 = 452
2: 2025 =49+51+...+99+101 ≡ 49⊕101 ≡ 27⊗75 = 512−242
3: 2025 =57+59+...+103+105 ≡ 57⊕105 ≡ 25⊗81 = 532−282
4: 2025 =121+123+...+147+149 ≡ 121⊕149 ≡ 15⊗135 = 752−602
5: 2025 =217+219+...+231+233 ≡ 217⊕233 ≡ 9⊗225 = 1172−1082
6: 2025 =401+403+405+407+409 ≡ 401⊕409 ≡ 5⊗405 = 2052−2002
7: 2025 =673+675+677 ≡ 673⊕677 ≡ 3⊗675 = 3392−3362
8: 2025 =2025 ≡ 2025 ≡ 1⊗2025 = 10132−10122
- 2025 is the sum of the first n primes that are of the form 4k+1.
The list of primes with this form begins 5, 13, 17, 29, ... .
What is n if the sum of 2025?Answer
n=21: 2025= 5+13+17+29+37+41+53+61+73+89+97+101+109+113+137+149+157+173+181+193+197
For other n, see A038346
Early Bird Numbers
We write out all the numbers in order as a single list of digits one after the other with no gaps, called the Natural Number Sequence. Clearly all numberswill eventually appear but some numbers appear before their "natural" place in the sequence, such as 21 which is the end of 12 and the start of the next number 13.
These are called Early Bird numbers.
For more on the Natural Number Sequence and Early Bird numbers, see An Introduction to the Natural Number Sequence 0123456789101112...There are Early Bird numbers in other concatenated sequence of digits, such as using the binary representatons of each natural number, or another base.
-
Where does 2025 first appear in the Natural Number sequence: in its "natural" place or before?
Answer
As a sequence of the decimal digits, 2025 first appears in its "natural" position ... 2024 2025 2026 ... (positions 20229..20239) - What if we write the numbers out in order with no gaps, as before,
but in binary?
The Binary Number Sequence starts (without the gaps): 0 1 10 11 100 101 110 ... .
Where does the binary for 2025 = 111111010012 first appear now?Answer
2025 in binary first appears at positions 1696..1706 in numbers 243 244 245: 11110011 11110100 11110101
Clock puzzles
-
20:25At 8:25pm a digital 24-hour clock shows 20:25. Ignoring the colon (:), which other square numbers will it show every day?
Answer
0, 1, 4, 9, 16, 25, 36, 49, 100, 121, 144, 225, 256, 324, 400, 441, 529, 625, 729, 841, 900, 1024, 1156, 1225, 1444, 1521, 1600, 1849, 1936, 2025, 2116, 2209, 2304
33 in all. see A122541
-
00:20:25If we had a digital clock showing seconds also, then both 00:20:25 = 2025 = 45² and 20:25:00 = 202500 = 450² are square numbers. How many times will it show a 6-digit square number now and what is the largest?
Answer
There are 217 squares shown on the 6-digit 24-hour clock, from 00:00:00 = 0² up to 23:52:25 = 485²
In Mathematica we haveClear[IsClockSec]; (* n can be seen on a digital 24 hour clock as hhmmss *) IsClockSec[n_] := Block[{digs = IntegerDigits[n]}, digs = Join[Table[0, 6 - Length[digs]], digs]; MemberQ[{0, 1, 2}, digs[[1]]] && If[digs[[1]] == 2, MemberQ[{0, 1, 2, 3}, digs[[2]]], True] && MemberQ[{0, 1, 2, 3, 4, 5}, digs[[3]]] && MemberQ[{0, 1, 2, 3, 4, 5}, digs[[5]]] ]; Select[Table[n -> n^2, {n, 0, Sqrt[240000]}], IsClockSec[#[[2]]] &] Length[%]
© 2024 Created: 24 December 2024 Dr Ron Knott
Home page - How many rectangles of any shape (including squares) can you draw using the lines of a grid which is n squares by n squares? Try it for 1 square, a grid of 2x2, of 3x3.