Gauss Technique
Write the sum a second way and the messy middle disappears.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: Write the sum a second way so the messy middle disappears.
L0 · Pair-and-Equalise (even n)
A helper is decorating thank-you cards for newcomer families. The first card gets 1 sticker. The second card gets 2 stickers. And so on, all the way to 8 stickers on the eighth card. How many stickers does the helper use in all?
L1 · Pair-and-Equalise (general step)
Volunteers are stacking cans at a food drive for newcomer families. Six rows hold 7, 9, 11, 13, 15, and 17 cans. How many cans are there altogether?
L2 · Gauss Pairing (odd n)
An animal rescue shelter has seven shelves of folded towels. The shelves hold 11, 14, 17, 20, 23, 26, and 29 towels. How many towels in all?
L3 · Recover n First
A neighbourhood support group packs lunch bags for kids in need. On the first day they pack 14 bags. Each day after that, they pack 3 more bags than the day before. On the last day they pack 47 bags. How many lunch bags do they pack in total?
L4 · Slice = Big Sum − Small Sum
In an encouragement-card campaign, volunteers send cards every day. Day 1: 1 card. Day 2: 2 cards. And so on, k cards on day k. How many cards do they send from day 38 through day 60?
L5 · Repeating Block Invariant
A community pantry tracks its daily net change in supply boxes. Three restocking days the pantry gains 1, 3, and 5 boxes. The next three days it gives out 7, 9, and 11. Then it gains 13, 15, and 17. Then it loses 19, 21, and 23. The pattern continues. What is the pantry's net change after the first 60 days?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
L0 · Stickers on Eight Cards
Peek the trick — Pair the smallest with the biggest
Pair the first and last terms; every outer pair adds to the same total. Multiply that pair-sum by the number of pairs.
L1 · Six Rows of Cans
Peek the trick — Even count, every term has a partner
When the number of evenly-spaced terms is even, every term has a perfect partner with the same pair-sum. Half-count of terms times pair-sum gives the total.
L2 · Seven Shelves of Towels
Peek the trick — Outsides pair, middle stands alone
When the number of evenly-spaced terms is odd, pair the outsides as usual and add the lone middle term. The Gauss formula still works directly.
L3 · Lunch Bags Every Day
Peek the trick — Find n before you sum
When the term count is hidden, recover it: n equals (last minus first) divided by step, plus one. Then plug n into the pair-and-equalise formula.
L4 · Cards from Day 38 to Day 60
Peek the trick — Rewrite the slice as a difference
A slice of a longer pattern equals the sum up to the end day minus the sum up to the day before the start. Two Gauss sums, one subtraction.
L5 · Pantry Net Change
Peek the trick — Sum one block, multiply by count
When the pattern repeats with a fixed-size signed block, total equals one block-sum times how many blocks fit. Spot the block, sum it once, multiply.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
