Why Odd Numbers Build Squares

A square grows on two sides

Begin with one tile. It is a one-by-one square: one row, one column, one unit of area. We will enlarge the square by increasing its side length one unit at a time. Watch the arriving tiles, because the size of that new border will explain the pattern in the totals.

A two-by-two square contains four tiles. Three new tiles surround the original one in an L-shaped border. One row has two tiles, and the remaining side contributes one more. The corner is already part of that row, so it is counted once. One plus three gives four.

A three-by-three square contains nine tiles. This time the new border adds five: three along one row and two more along the remaining side. The previous four-tile square remains inside the enlarged arrangement. The addition changes the side length from two to three and the area from four to nine.

Come closer to the corner of this square. Follow the row of three, then the two remaining tiles along the other side. If you counted three in both directions, you would include the corner twice. The border has five different tiles. Its L shape keeps the shared corner visible.

The next border is larger

Here is the complete three-by-three square again. Its total is nine, and its side length is three tile widths. Those quantities are connected by three times three. To reach the next square, we need room for a fourth row and a fourth column, sharing one new corner.

The four-by-four square arrives, containing sixteen tiles. Its new border contributes seven: a row of four plus three further tiles along the side. Nine plus seven gives sixteen. The border is larger than the preceding five-tile border because both arms have grown by one.

Follow the new row of four. This row includes the corner where the two arms meet. Once those four are counted, only three tiles remain along the other arm. That is one way to organize the seven without recounting a tile. You could reverse which arm you count first and reach the same result.

Return to the whole square. Compare the side length four with the area sixteen. Adding one to the side did not add one to the area. It added the entire L-shaped border. Whenever a length changes, the associated area needs its own accounting; they measure different features.

Why each border is odd

A five-by-five square contains twenty-five tiles. Its new border adds nine to sixteen. Count five across one arm, then four more along the other. The two arms differ in length by one when the corner belongs to the first arm. That difference is the source of the odd total.

Look near the corner again. Pair four positions along one arm with four along the other in your attention, leaving the corner as the extra tile. We are describing a pairing of the visible positions, not moving them into a different arrangement. Four pairs and one extra make nine.

A six-by-six square contains thirty-six tiles. Its border adds eleven: six along a row and five along the remaining side. Five pairs and one extra give the same count. The border pattern has continued through three, five, seven, nine and eleven, each two larger than the previous contribution.

The complete six-by-six square stays in view. Every step adds one tile to the length of each arm of the next border, so the border grows by two. Starting with an odd contribution and adding two keeps the contributions odd. The geometric change explains the pattern instead of merely naming it.

Odd contributions accumulate into squares

Return to the single tile. It is the starting contribution of one. The larger squares can be read as this starting tile together with all the borders added afterward. We will rebuild a few steps, keeping track of the accumulating contributions rather than only the side lengths.

Add the three-tile border and the total becomes four. The accumulated expression is one plus three. Its result is a square because the new border fills exactly the row and column needed for the next side length. No tile is left outside the square and no corner is missing.

Add the five-tile border and the total becomes nine. One plus three plus five now describes the complete three-by-three square. The last contribution is five, but the total includes the earlier four as well. Distinguishing the newest part from the whole prevents the two sequences from being confused.

Add seven and the total becomes sixteen. One plus three plus five plus seven is four squared. There are four odd contributions, including the starting one, and the resulting square has side length four. The count of contributions and the side length agree here, while the area is sixteen.

Connect the square with neighboring triangles

A three-row triangle contains six units. We will compare it with the next triangle and then with a square. These views appear one at a time, so keep each total in mind. The comparison connects two families we have already seen; it does not turn six units into sixteen without adding anything.

The four-row triangle contains ten. Six and ten together make sixteen. Their row counts differ by one. If a four-by-four square is divided along a diagonal of tiles, one side can include that diagonal while the other excludes it. The two groups then have these neighboring triangular counts.

Here is the four-by-four square of sixteen. The earlier triangles were shown separately, rather than laid over this square. Look along a diagonal and mentally assign its tiles to one of the two sides. That avoids counting the diagonal twice. The comparison explains how neighboring triangular totals can account for a square total.

Move closer to the square, then return to its complete outline. The row-and-column description gives four times four. The growing-border description gives one plus three plus five plus seven. The neighboring-triangle comparison gives six plus ten. These different expressions describe the same sixteen units for different reasons.

Check the rule across new examples

Begin with the seven-by-seven square, area forty-nine. Predict the border needed for side eight. One complete new row will contain eight tiles, and seven further tiles will complete the other side. The shared corner belongs to the row already counted. Eight plus seven gives fifteen, so the predicted new area is sixty-four.

The eight-by-eight square now appears, with its fifteen-tile border surrounding the old forty-nine. Inspect the added row and the remaining arm. Counting eight in both directions would count sixteen and include the shared corner twice. The correct count is eight plus eight minus one, which equals the same fifteen.

Approach the meeting corner. The corner is one actual tile, even though it lies in both the new row and the new column. Its membership in two named groups does not make two copies of it. This is why counting overlapping groups requires a correction, and why assigning the corner to only one arm avoids that correction from the start.

Return to the full square and grow its side from eight to nine. The new border contributes nine plus eight, or seventeen. The area becomes eighty-one. Compare the two successive borders: fifteen then seventeen. Increasing the side again lengthens both arms, so the next border has two more tiles than the preceding one.

Read the complete nine-square. Relative to the original seven-square, it contains two new borders, not only the outer seventeen. Fifteen plus seventeen equals thirty-two, and eighty-one minus forty-nine also equals thirty-two. Jumping two side lengths requires accounting for both stages of growth; the final border alone describes only the last stage.

Return to the seven-square to check the difference in reverse. Its forty-nine tiles are thirty-two fewer than the eighty-one in the nine-square. We have compared complete arrangements in succession, rather than claiming that the camera removed particular tiles. The numerical difference agrees with the two visible growth contributions we followed, fifteen and seventeen.

Predict a border and its new total

Return to the six-by-six square containing thirty-six units. Predict the next border before it appears. A new row will contain seven tiles; the remaining side will contain six. Together they will contribute thirteen. This prediction uses the two arms, rather than guessing from the previous total alone.

The seven-by-seven square now contains forty-nine tiles. The thirteen-tile border has arrived around the old thirty-six. Six pairs plus one extra give thirteen, and thirty-six plus thirteen gives forty-nine. Check the new side length and the complete area separately: seven and forty-nine answer different questions.

Follow the row of seven. Once its corner is included, six tiles remain on the other arm. For a square growing from side n to side n plus one, the same counting pattern gives n plus one along a row and n along the rest of the side. Together that is two n plus one.

Here is the complete square again. The expression two n plus one records the border we just counted; it comes after the visible relationship. You can now move between a side length, a square total and the next odd contribution. Try another modest square and explain its next border before advancing to see it.