Five Options, One Blank Cell
A 3-by-3 matrix of shapes is printed on a card with the bottom-right cell left blank; five options are offered for it. The eight visible cells change in two ways at once.
Across each row, the outer shapes always run triangle, square, pentagon - thin black outline, white interior, all the same size and all upright. Row 2 repeats that exact order. Row 3 begins with a triangle, then a square, then the blank cell.
Down the columns, only the contents change. Every shape in row 1 is empty. Every shape in row 2 holds one small solid black dot at its centre. Every shape in row 3 holds two small solid black dots, placed side by side on a horizontal line.
Nothing else varies: no rotation, no resizing, no outline change.
The five options show: a pentagon with three dots; a square with two dots; a pentagon with two dots; a pentagon with two hollow circles; a triangle with two dots.
1. State every attribute the blank cell must have, and name the specific visible cells that establish each one.
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Three attributes, each with its own evidence:
- Outer shape - pentagon. The shape is set by the column, not the row: the first cell of row 3 is a triangle and the second is a square, so the third must be a pentagon. Rows 1 and 2 confirm the same column order independently.
- Number of dots - two. The count is set by the row. Both filled cells in row 3 already show two dots, so the blank must match its neighbours.
- Dot style - solid black. Rows 1 and 2 show solid dots only, and nothing in the matrix alternates shading.
Also fixed by the matrix: upright, same size, thin black outline, white interior, and the two dots stay side by side on a horizontal line.
Answer: a pentagon containing two solid black dots, upright.
2. One option shows a pentagon with three dots. What reasoning error produces it, and what single check rules it out?
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The error is reading the dot count as a running total - one more than the cell to its left, or a diagonal sweep from the top-left - instead of as a row constant.
The one-step check: look at the other two cells in row 3. Both already hold exactly two dots, so the count cannot climb along that row. Column 3 gives no support either; its members are empty, one dot, and unknown, which is not a sequence of any kind.
The general test: a rule must fit every visible cell it claims to explain. The running-total reading fits none of the two visible cells in row 3, so it fails before the blank cell is even considered.
3. Suppose a colleague insists the finished matrix must also obey a diagonal rule. Test that claim against the actual cells and say whether your answer changes.
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Main diagonal cells: top-left is a triangle with no dots, centre is a square with one dot, bottom-right would be the pentagon with two dots. Those three share no property except position - different shapes, different dot counts - so no diagonal rule is available, and the answer stands unchanged.
There is a subtler point worth making. A diagonal reading that is consistent - dots equal the column number minus one - reproduces exactly the same counts as the row rule. It is supported, but it is also redundant: it predicts nothing the row rule did not already predict. That is the useful distinction for a matrix item. An extra rule must either be contradicted by a visible cell (discard it) or add a prediction the existing rules miss (keep it). Here it does neither, so the answer stays a pentagon with two solid dots.