Problem
Find the rule in stacked equations
A staircase reads 5 × 5, 15 × 15, 25 × 25, 35 × 35. The dots mean the list keeps going the same way. Work out the rule that turns each number into its answer. Use the rule to find 495 × 495.
Reasoning · Grade 4-1 Number Arrangement Tables
Your answer
Watch out The front looks like the tens digit squared, but it is that digit times the next one.
How to solve
Strategy Look for a Pattern — The four answers 25, 225, 625 and 1225 look unrelated until they are cut into two pieces. Every one of them ends in 25, so I split each answer into 'the part in front' and 'the 25 at the back' and line the front parts up against the numbers being squared. Once the front parts are read as a small multiplication, the rule appears, and I can test it on a line not printed in the problem before trusting it on 495.
1STEP 1
Notice that every answer ends in 25
Every answer ends in 25.
25, 2|25, 6|25, 12|25
Splitting each answer at the same place turns four different-looking numbers into one fixed part plus one changing part, and only the changing part still needs explaining.
4.NBT.A.2Organize Information In More Ways2STEP 2
Line the front parts up against the tens digits
The fronts 0, 2, 6, 12 track the tens digit.
0 → 0, 1 → 2, 2 → 6, 3 → 12
Putting the input and the output side by side in a table is what makes a hidden rule visible.
4.OA.C.5Organize Information In More Ways3STEP 3
Read the front parts as a multiplication
The front is that number times the next.
0 × 1 = 0, 1 × 2 = 2, 2 × 3 = 6, 3 × 4 = 12
Two consecutive whole numbers multiplied together is one of the first patterns young solvers meet, so once the outputs are written as 0, 2, 6, 12 the rule is recognisable on sight.
4.OA.C.5Look For A Pattern4STEP 4
Test the rule on a line that was not printed
Testing on the unprinted 45 × 45 gives 2025.
45 × 45 = 1800 + 225 = 2025
A rule spotted from four examples is only a guess until it predicts a fifth one correctly, and 45 times 45 is small enough to check the long way.
4.NBT.B.5Solve An Easier Related ProblemA rule spotted from four printed lines stays a guess until it correctly predicts a line that was not printed.
Why?
One line where the rule gives the wrong answer would prove it false, so testing an unseen line is a real test and not a formality.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
The front parts climb by a pattern that grows in a steady way, so the rule has to keep working as the inputs march upward.
🧱A constant step reaches any termWhen a list climbs by the same amount every time, any term is the starting value plus that step repeated, so you never have to write out the ones between.
5STEP 5
Apply the rule to 495
49 × 50 = 2450, so the answer is 245025.
49 × 50 = 2450 → 495 × 495 = 245025
The rule replaced a three-digit by three-digit multiplication with one easy two-digit multiplication, and 49 times 50 is easy because 50 is half of 100.
4.NBT.B.5Look For A PatternAnswer
245025
49 × 50 = 2450
A rough size check: 495 is close to 500, and 500 times 500 is 250000, so the answer should be a little under 250000. The value 245025 is just under, so the magnitude is right. The last two digits are 25 as the rule requires, and the front part 2450 is exactly 49 times 50. A direct check also works: 495 times 495 is 495 times 500 minus 495 times 5, that is 247500 minus 2475, which is 245025.
Takeaway
Any number ending in 5, times itself, is just the front part times the next number with 25 stuck on the end.
- Notice that every answer ends in 25
- Line the front parts up against the tens digits
- Read the front parts as a multiplication
- Test the rule on a line that was not printed
- Apply the rule to 495