Why a pattern always holds
Letter-numbers cover every case at once.
A calendar puzzle
Pick a 2 × 2 square in a calendar, like 12, 13, 19, 20. One diagonal gives 12 + 20 = 32. The other gives 13 + 19 = 32. They are equal!
Imagine the calendar goes on with endless rows. Will the two diagonal sums be equal in every 2 × 2 square?
What this lesson covers
The idea
Checking some cases cannot show that a pattern always holds; writing a general case with letter-numbers and simplifying shows it holds for every value.
A calendar puzzle
Pick a 2 × 2 square in a calendar, like 12, 13, 19, 20. One diagonal gives 12 + 20 = 32. The other gives 13 + 19 = 32. They are equal!
Imagine the calendar goes on with endless rows. Will the two diagonal sums be equal in every 2 × 2 square?
- Yes, in every square
- Only in some squares
- We can only know by testing every square
Try it, then prove it
Tap the calendar and try a few squares. Then call the top-left number a letter-number and see what happens.
One letter-number covers every square
We cannot test endless squares. But every square is described by its top-left number, whatever it is. Call it a. The number to its right is 1 more, the one below is 7 more, and the diagonal one is 8 more: a, a + 1, a + 7, a + 8.
Both diagonal sums are 2a + 8, whatever a is. So they are equal in every square, not only the ones we tried.
Checking some cases cannot show that a pattern always holds. Write the general case with letter-numbers and simplify: if both sides simplify to the same expression, the pattern holds for every value.
- Diagonal | Sum | Simplified
- a and a + 8 | a + (a + 8) | 2a + 8
- a + 1 and a + 7 | (a + 1) + (a + 7) | 2a + 8
Notes
Checking some cases cannot show that a pattern always holds. Writing the general case with letter-numbers and simplifying shows it holds for every value: both diagonal sums are 2a + 8.
Check yourself
Ria tries 10 different 2 × 2 squares. The diagonal sums are equal every time. Has she shown this holds for every square?
The diagonal sum of a 2 × 2 square is 2a + 8. Find it for the square whose top-left number is 50.
Answer: 108
2 × 50 + 8 = 108. Check: 50 + 58 = 108 and 51 + 57 = 108.
In a calendar cross the centre is c. The other four numbers are c − 7, c − 1, c + 1 and c + 7. What does the sum of all five numbers simplify to?
The five numbers of a calendar cross add up to 100. What is the centre number?
Answer: 20
5c = 100, so c = 20. The cross is 13, 19, 20, 21, 27: 13 + 19 + 20 + 21 + 27 = 100.
- Yes: it worked 10 times, so it always works.. Ten cases are only ten cases. Another square might break the pattern.
- Yes, if she tries 100 squares instead.. However many she tests, endless squares are still left.
- No: there are endless squares. A general case with a letter-number covers all of them. — correct. Yes! Testing cases cannot cover endless squares.
- c + 5. There are five c’s, so the sum is 5 times c, not c plus 5.
- 4c. The four neighbours give 4c, but the centre c is in the sum too.
- 5c — correct. Yes! The five c’s give 5c, and −7 − 1 + 1 + 7 = 0.