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27 Nov 2024, 06:29
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Must be true: An unpolished garnet is included in the display.
Must be false: Exactly two polished garnets are included in the display.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
73% (02:55) correct
27%
(02:52)
wrong
based on 746
sessions
History
Date
Time
Result
Not Attempted Yet
Three types of gemstones—emeralds, garnets, and amethysts—discovered during a recent archaeological expedition will be selected for display at a local museum based on the following criteria:
• Six stones will be chosen for the display, including both polished and unpolished stones.
• No more than three stones of any one type will be included, and at least one stone of each type must be selected.
• At least one unpolished garnet must be included.
• If more than one emerald is selected, the same number of amethysts, with the same finish (polished / unpolished), must also be selected.
If two polished emeralds are selected for the display, and assuming that all the criteria are met, select for Must be true the statement that must be true, and select for Must be false the statement that must be false. Make only two selections, one in each column.
• Six stones will be chosen for the display, including both polished and unpolished stones.
• No more than three stones of any one type will be included, and at least one stone of each type must be selected.
• At least one unpolished garnet must be included.
• If more than one emerald is selected, the same number of amethysts, with the same finish (polished / unpolished), must also be selected.
If two polished emeralds are selected for the display, and assuming that all the criteria are met, select for Must be true the statement that must be true, and select for Must be false the statement that must be false. Make only two selections, one in each column.
| Must be true | Must be false | |
| An unpolished garnet is included in the display. | ||
| A polished garnet is included in the display. | ||
| Exactly two unpolished garnets are included in the display. | ||
| Exactly two polished garnets are included in the display. | ||
| Both a polished garnet and an unpolished garnet are included in the display. |
ShowHide Answer
Official Answer
Must be true: An unpolished garnet is included in the display.
Must be false: Exactly two polished garnets are included in the display.
01 Dec 2024, 21:30
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Bunuel
Official Solution:
Since more than one emerald is selected (specifically two polished emeralds), the criteria require that both the number and type of emeralds and amethysts must match. Thus, two polished amethysts must also be included. Selecting three emeralds and three amethysts would leave no room for a garnet, which violates the rule that at least one of each type must be included. Therefore, the display includes exactly two emeralds and exactly two amethysts, all polished.
This leaves two remaining stones, which must both be garnets, with at least one of them unpolished to satisfy the criteria. Thus:
• Must be true: An unpolished garnet is included in the display.
• Must be false: Exactly two polished garnets are included in the display.
Correct answer:
Must be true "An unpolished garnet is included in the display. "
Must be false "Exactly two polished garnets are included in the display."
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General Discussion
10 Feb 2026, 21:53
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Let's break this down systematically.
Step 1: List all the rules
• 6 stones total for display
• Max 3 of any one type; at least 1 of each type (emeralds, garnets, amethysts)
• At least 1 unpolished garnet (critical constraint!)
• If more than 1 emerald → same number of amethysts with same finish
Step 2: Apply the given condition
We're told: 2 polished emeralds are selected.
Since 2 emeralds > 1 emerald, the fourth rule kicks in:
→ We must have 2 polished amethysts (same number, same finish)
Step 3: Count what we have
• 2 polished emeralds + 2 polished amethysts = 4 stones used
• Remaining slots: 6 - 4 = 2 stones
Step 4: What fills the remaining 2 slots?
• We already have emeralds and amethysts
• We need garnets (at least 1 of each type rule)
• Therefore: Both remaining slots go to garnets
Step 5: What type of garnets?
• We must have at least 1 unpolished garnet (third rule)
• The second garnet can be polished OR unpolished
Possible scenarios:
✓ Scenario A: 2 unpolished garnets
✓ Scenario B: 1 unpolished + 1 polished garnet
Step 6: Evaluate each statement
(A) An unpolished garnet is included
→ MUST BE TRUE – the third rule requires at least 1 unpolished garnet. This is always true.
(B) A polished garnet is included
→ Could be true (Scenario B) or false (Scenario A). Not definite.
(C) Exactly 2 unpolished garnets are included
→ Only true in Scenario A, not in B. Not definite.
(D) Exactly 2 polished garnets are included
→ MUST BE FALSE – We only have 2 garnet slots, and at least 1 must be unpolished. Having 2 polished garnets is impossible.
(E) Both a polished and unpolished garnet are included
→ Only true in Scenario B, not in A. Not definite.
Answer: Must be True = A | Must be False = D
Step 1: List all the rules
• 6 stones total for display
• Max 3 of any one type; at least 1 of each type (emeralds, garnets, amethysts)
• At least 1 unpolished garnet (critical constraint!)
• If more than 1 emerald → same number of amethysts with same finish
Step 2: Apply the given condition
We're told: 2 polished emeralds are selected.
Since 2 emeralds > 1 emerald, the fourth rule kicks in:
→ We must have 2 polished amethysts (same number, same finish)
Step 3: Count what we have
• 2 polished emeralds + 2 polished amethysts = 4 stones used
• Remaining slots: 6 - 4 = 2 stones
Step 4: What fills the remaining 2 slots?
• We already have emeralds and amethysts
• We need garnets (at least 1 of each type rule)
• Therefore: Both remaining slots go to garnets
Step 5: What type of garnets?
• We must have at least 1 unpolished garnet (third rule)
• The second garnet can be polished OR unpolished
Possible scenarios:
✓ Scenario A: 2 unpolished garnets
✓ Scenario B: 1 unpolished + 1 polished garnet
Step 6: Evaluate each statement
(A) An unpolished garnet is included
→ MUST BE TRUE – the third rule requires at least 1 unpolished garnet. This is always true.
(B) A polished garnet is included
→ Could be true (Scenario B) or false (Scenario A). Not definite.
(C) Exactly 2 unpolished garnets are included
→ Only true in Scenario A, not in B. Not definite.
(D) Exactly 2 polished garnets are included
→ MUST BE FALSE – We only have 2 garnet slots, and at least 1 must be unpolished. Having 2 polished garnets is impossible.
(E) Both a polished and unpolished garnet are included
→ Only true in Scenario B, not in A. Not definite.
Answer: Must be True = A | Must be False = D
