Certainly! To address your task, I will:

1. **Identify appropriate quantitative physicochemical properties** for classifying these BASic organic compounds.
2. **Gather representative data** (from standard property databases/literature) for each compound, focusing on quantitative measures: primarily, **pKa of the conjugate acid** (which quantitatively measures basicity in organic bases) and, where relevant, properties such as **steric bulk** (e.g., calculated van der Waals volume), or **H-bond donors** (for possible H-bonding).
3. **Define clear, database-like categories** for each property (e.g., "Strong base" if pKa > 12, etc.), and assign each substance.
4. **Conclude whether the three belong together or should be separated—based solely on property data.**

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### 1. **Selection of Key Quantitative Physicochemical Properties for Strong Bases**

- **Basicity**: **pKaH** (pKa of the conjugate acid in water/acetonitrile; higher = stronger base)
- **Steric Bulk**: E.g., number of non-hydrogen atoms or calculated van der Waals volume (if relevant to basicity/solubility)
- **Hydrogen Bond Donors**: Count (usually 0 for strong non-nucleophilic bases)
- **Polarity**: Often expressed as logP or dipole moment, but not primary for SIMPLE base classification.

For "Base" classification, the **pKa of the conjugate acid** is universally the leading quantitative property, and this should be our principal criterion.

---

### 2. **Compilation of pKaH Values** (Quantitative Data)

#### **1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine**

- Structure: Bicyclic amidine, methylated, hexahydro.
- **pKaH** (Conjugate acid): This class (e.g., DBU) has **pKaH ≈ 13.5 (water)**, up to 24 in acetonitrile.

#### **2-tert-butyl-1,1,3,3-tetramethylguanidine**

- Structure: One of the "proton sponges" in the guanidine family.
- **pKaH**: Closest analogues (e.g., 1,1,3,3-tetramethylguanidine, TMG) have **pKaH = 13.6 (water)**; with tert-butyl, this could be 14–15.

#### **N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine**  
(= **P1-t-Bu [Phosphazene base, family of very strong non-nucleophilic bases]**)

- Structure: Polyaminophosphazene backbone; one of the strongest known organic bases.
- **pKaH**: **~29.5 (acetonitrile)**; ~18–19 (water).
- **References:** Schwesinger, R. et al, JCS Chem. Commun. 1987, 892–894

---

### 3. **Defining Classification Intervals**

Let us define **three groups** (maximum), based on **pKa of conjugate acid in water** (since all three are strong organic superbase candidates):

| Class | pKaH (water)       | pKaH (MeCN; typical for strong bases) | Label                       |
|-------|---------------------|---------------------------------------|-----------------------------|
| A     | pKaH < 14           | < 18                                  | "Conventional strong base"  |
| B     | 14 ≤ pKaH < 19      | 18–25                                 | "Superbase" (enhanced)      |
| C     | pKaH ≥ 19           | > 25                                  | "Extremely strong superbase" (Phosphazene class) |

**Note:** Many physicochemical handbooks use pKaH values in water for direct comparability, but base strength in organic synthesis is best compared using acetonitrile values, which are much higher and differentiate superbase classes.

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### 4. **Applying the Classification**

#### (a) **1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine**
- **pKaH:** ~13.5 (water); ~24 (MeCN)
- **Category:** *Conventional strong base* (A/B border)

#### (b) **2-tert-butyl-1,1,3,3-tetramethylguanidine**
- **pKaH:** ~14–15 (water); ~24–25 (MeCN)
- **Category:** *Superbase* (B)

#### (c) **N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine** *(Phosphazene base)*
- **pKaH:** ~18–19 (water); ~29.5 (MeCN)
- **Category:** *Extremely strong superbase (Phosphazene)* (C)

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### 5. **Justification for Number of Classes**

- All are strong organic bases, but **phosphazene base** is >10⁵ times stronger (by pKa) than the other two.
- There is at least an order of magnitude difference in base strength (quantitatively, in pKa units) between the DBU/guanidine-type and the phosphazene base.
- The **DBU derivative's** and **guanidine's** pKaH are extremely close, and could (if preferred) potentially be put in a single "strong base" group separate from phosphazenes.
- Strict adherence to "NO MORE THAN THREE GROUPS" can be met easily; three is justified by huge jumps in basicity.

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## **Final Classification Table** (with Quantitative Data)

| Compound                                                                             | pKaH (water)        | pKaH (MeCN)     | Hydrogen bond donors | Class         | Notes                          |
|--------------------------------------------------------------------------------------|---------------------|-----------------|---------------------|---------------|--------------------------------|
| 1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine                              | ~13.5               | ~24             | 0                   | A             | DBU-type bicyclic amidine      |
| 2-tert-butyl-1,1,3,3-tetramethylguanidine                                            | ~14–15              | ~24–25          | 0                   | B             | Sterically hindered guanidine  |
| N-[dimethylamino-ethylimino-[[tris(dimethylamino)...phosphanyl]-N-methylmethanamine  | ~18–19              | ~29.5           | 0                   | C             | Phosphazene base               |

**Group Definitions—Quantitative:**

- **Group A: Conventional strong base** (pKaH < 14)
- **Group B: Superbase** (14 ≤ pKaH < 19)
- **Group C: Extremely strong superbase (Phosphazene base)** (pKaH ≥ 19; best exemplified by the phosphazenes)

---

### **Summary**

**Based strictly on available quantitative property data:**
- **1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine** and **2-tert-butyl-1,1,3,3-tetramethylguanidine** are both strong organic bases (pKaH ~13.5–15) and could be grouped together (if you wish to use only two categories);
- **N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine** is quantitatively a much stronger base (pKaH ~19, MeCN ~30), and must be grouped separately per the established classification based on pKaH.

**Therefore: The most correct, quantitatively justified classification is THREE groups, primarily based on pKa of the conjugate acid.**  
If only two categories were allowed, the first two could be lumped as "strong bases," and the phosphazene stands alone as a "superbase."

---

## **Final Answer Table**

| Group   | Compound(s)                                                                                                       | Criteria (Quantitative)                       |
|---------|-------------------------------------------------------------------------------------------------------------------|-----------------------------------------------|
| Group 1 | 1-methyl-2,3,4,6,7,8-hexahydropyrimido[1,2-a]pyrimidine, 2-tert-butyl-1,1,3,3-tetramethylguanidine                | pKaH (water) < 16; pKaH (MeCN) < 26          |
| Group 2 | N-[dimethylamino-ethylimino-[[tris(dimethylamino)-lambda5-phosphanylidene]amino]-lambda5-phosphanyl]-N-methylmethanamine (Phosphazene base) | pKaH (water) ≥ 16; pKaH (MeCN) ≥ 26          |

**If permitted, ALL could be grouped as "very strong bases" compared to e.g., amines (pKa below 11), but the phosphazene is >10⁵ times more basic, so this would gloss over a crucial distinction. Quantitatively, two groups are most justified, but three are possible if you wish to subdivide the first two (as shown above).**

**CONCLUSION:**  
***Quantitative pKa values directly support placing the phosphazene base in its own (superbase) class, while the DBU-type and guanidine-type derivatives can reliably be grouped together as strong organic bases.***