CEMC Pascal, 2014

Algebra Difficulty 2.5 Junior Find the answer Canada

In the $4\times 4$ grid shown, each of the four symbols has a different value. The sum of the values of the symbols in each row is given to the right of that row.

What is the value of $\blacklozenge$ ?

Pick one

Solutions — 2

Solution 1

From the second row, $\triangle + \triangle + \triangle + \triangle = 24$ or $4\triangle = 24$ , and so $\triangle = 6$ .

From the first row, $\heartsuit + \triangle + \triangle + \heartsuit = 26$ or $2 \heartsuit + 2 \triangle = 26$ .

Since $\triangle = 6$ , then $2 \heartsuit = 26 - 12 = 14$ , and so $\heartsuit = 7$ .

From the fourth row, $\square + \heartsuit + \square + \triangle = 33$ .

Since $\triangle = 6$ and $\heartsuit = 7$ , then $2\square + 7 + 6 = 33$ , and so $2\square = 20$ or $\square = 10$ .

Finally, from the third row, $\square +$ $\blacklozenge +$ $\heartsuit +$ $\blacklozenge$ $=27$ . Since $\square = 10$ and $\heartsuit = 7$ , then $2$ $\blacklozenge$ $= 27 - 10 - 7 = 10$ .

Thus, $\blacklozenge = 5$ .

Solution 2

From the second row, $\triangle + \triangle + \triangle + \triangle = 24$ or $4\triangle = 24$ , and so $\triangle = 6$ .

From the first row, $\heartsuit + \triangle + \triangle + \heartsuit = 26$ or $2 \heartsuit + 2 \triangle = 26$ .

Since $\triangle = 6$ , then $2 \heartsuit = 26 - 12 = 14$ , and so $\heartsuit = 7$ .

From the fourth row, $\square + \heartsuit + \square + \triangle = 33$ .

Since $\triangle = 6$ and $\heartsuit = 7$ , then $2\square + 7 + 6 = 33$ , and so $2\square = 20$ or $\square = 10$ .

Finally, from the third row, $\square +$ $\blacklozenge$ $+ \heartsuit +$ $\blacklozenge$ $= 27$ .

Since $\square = 10$ and $\heartsuit = 7$ , then $2$ $\blacklozenge$ $= 27 - 10 - 7 = 10$ .

Thus, $\blacklozenge$ $= 5$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.