Problem
Who sits opposite in a circle
Twenty-two students numbered 1 to 22 sit in a circle. They are evenly spaced and in numerical order. Opposite means straight across the middle. Find who sits opposite number 8.
Reasoning · Grade 5-1 Rules and Correspondence
Your answer
Watch out Adding 11 usually works, but past the end you must subtract 22.
How to solve
Strategy Draw a Diagram — A quick sketch of the ring shows the key fact straight away: two students facing each other cut the circle into two halves that must hold the same number of people. Rather than drawing all 22 seats, I try an easier circle first — six students, then eight — find the rule connecting facing numbers, then check that the rule still works when the count goes past 22 and wraps back to 1.
1STEP 1
Draw the ring and see what "across" means
Removing the pair leaves 10 on each side.
22 - 2 = 20 students left, 20 ÷ 2 = 10 on each side
Once the picture is on paper the equal-halves fact is something you can see rather than something you have to be told; evenly spaced seats mean the two arcs are mirror images.
4.OA.A.3Draw A Diagram2STEP 2
Test the rule on a smaller circle
A small circle shows they sit half the ring apart.
6 ÷ 2 = 3: 1 ⇔ 4, 2 ⇔ 5, 3 ⇔ 6
A circle of six is small enough to draw completely and check every pair by eye, and whatever rule holds there must hold for 22 too, because nothing about the reasoning depended on the number six.
4.OA.C.5Solve An Easier Related Problem3STEP 3
State the rule for 22 students
For 22 students half is 11.
22 ÷ 2 = 11
Half the circle is half the seats, and since the numbers go up by one per seat, going halfway round adds half of 22 to the number.
4.OA.C.5Look For A PatternWith 22 students, the seat across the circle is found by adding 11 and wrapping back round if the number passes 22.
Why?
The seats form a closed ring, so stepping off the end simply brings you back in at the beginning.
🧱A cycle returns after a full periodA fixed repeating cycle of length p comes back to the same position every p steps.
Why?
Adding a whole lap of 22 changes nothing, so only what is left after whole laps decides which seat you land on.
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
4STEP 4
Apply the rule to number 8
Adding 11 to 8 gives 19.
8 + 11 = 19
This is a single addition once the diagram has told you what to add, which is the whole point of drawing it first.
4.OA.A.3Draw A Diagram5STEP 5
Check the wrap-around, both ways round
Counting the other way also gives 19.
8 - 11 + 22 = 19, 19 + 11 - 22 = 8
On a circle, adding or subtracting the full 22 brings you back to the same seat, so it is always safe to add 22 to fix a number that has slipped below 1 or subtract 22 from one that has run past 22.
4.OA.C.5Make A Systematic ListAnswer
19
8 + 11 = 19
The answer must be one of the jersey numbers 1 to 22, and 19 is. It must also not be 8 itself or one of 8's neighbours, and 19 is far away round the ring, as a facing seat should be. Pairing everyone up confirms the count works out: 1⇔12, 2⇔13, …, 11⇔22 uses each of the 22 numbers exactly once and gives 11 facing pairs, and the pair containing 8 is 8 ⇔ 19. Note this only works because 22 is even; with an odd number of students nobody would sit directly opposite anyone.
Takeaway
Facing each other means splitting the circle in half, so the two jersey numbers differ by half the crowd: 22 / 2 = 11, and 8 + 11 = 19.
- Draw the ring and see what "across" means
- Test the rule on a smaller circle
- State the rule for 22 students
- Apply the rule to number 8
- Check the wrap-around, both ways round